Incidence Graphs and Unneighborly Polytopes - DepositOnce

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Incidence Graphs and
Unneighborly Polytopes
vorgelegt von
Diplom-Mathematiker
Ronald Frank Wotzlaw
aus Göttingen
Von der Fakultät II – Mathematik und Naturwissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
– Dr. rer. nat. –
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Fredi Tröltzsch
Berichter: Prof. Günter M. Ziegler
Prof. Gil Kalai
Tag der wissenschaftlichen Aussprache: 6. Februar 2009
Berlin 2009
D 83
. . . for though it cannot hope to be useful or informative on all matters,
it does at least make the reassuring claim, that where it is inaccurate
it is at least definitively inaccurate.
In cases of major discrepancy it’s always reality that’s got it wrong.
The Restaurant at the End of the Universe
Douglas Adams
For Finja and Nora
Preface
This thesis is about polytopes and “their life in high dimensions.” Nearly
all questions considered here are trivial or simple exercises when restricted
to dimensions that we can “see.” More often than not, the interesting
examples start to appear in dimensions d = 6, 7, 8 or even much higher.
Nevertheless, I have tried to illustrate ideas with pictures throughout. The
reader should be warned however that they rarely show the real nature of
the mathematical problem at hand.
Some of the results are generalized to objects that are more abstract than
polytopes. I have taken a pragmatical point of view with respect to the level
of generality that theorems are stated and proved: If a generalization was
asked for in the existing literature, it is provided (if possible).
The following is a list of the main results in this thesis in order of
appearance:
• The connectivity of (k, ℓ)-incidence graphs is determined for the case
ℓ ≥ 2k + 1 in Chapter 3.
• Athanasiadis’ conjecture on incidence graphs of Cohen-Macaulay cell
complexes is proved in Chapter 4.
• Linkages in polytopes on at most f0 ≤ (6d+7)/5 vertices are analyzed
completely in Chapter 5; this is based on joint work with Axel Werner.
• Perles’ Skeleton Theorem is proved and generalized in a number of
ways in Chapters 6, 7, and 8. In particular, this theorem is reestablished for graded relatively complemented lattices and for pyramidally perfect lattices.
• A complete solution to Mani’s problem on nonsimplicial illuminated
polytopes is obtained in Chapter 9.
• Counterexamples to Marcus’ conjecture on positive k-spanning sets
are constructed for all k ≥ 2 in Chapter 10.
• Bienia & Las Vergnas’ problem on bounding the size of positive kspanning sets in oriented matroids is solved in Chapter 11.
vii
Apart from these results, many small improvements are made to existing
results, and some unpublished results are re-established. The following is a
list of minor results (some of them only partially new):
• Counterexamples to simplex refinements of polytopes with certain
properties are given in Chapter 2. These examples are originally due
to Lockeberg, but our analysis of them is simpler.
• Counterexamples to a strong version of a conjecture about centrally
symmetric polytopes due to Grünbaum are given in Chapter 2.
• Two generalizations of Balinski’s theorem are combined in Chapter 3
to yield the so far strongest version of this theorem.
• A partial answer to a question by Athanasiadis on polytopality of
incidence graphs is given in Chapter 3.
• Larman & Mani’s lower bound on linkedness of general polytopes is
improved marginally in Chapter 5. This improvement implies exact
values for linkedness of d-polytopes for d ≤ 5 and for d = 7, 8, 10, 13.
This thesis was written in the time period between April 2006 and December 2008.
I had the enormous luck to be financially supported by the Deutsche
Forschungsgemeinschaft (DFG) via the Research Training Group Methods for Discrete Structures (MDS) and by the Berlin Mathematical School
(BMS). In particular, the additional support for students with children has
made my life a lot easier.
A number of people had their influence on this thesis, and I want to
mention them here.
First and foremost, I thank my advisor Günter M. Ziegler. Without
him, this thesis simply would not exist. He offered me a scholarship, first
on his Leibniz grant and later in the MDS and in the BMS. These scholarships made it possible to attend conferences around the world and to visit
Julian Pfeifle in Barcelona in the spring of 2008. Günter also helped me get
invited to the MSRI Program on Computational Applications of Algebraic
Topology in 2006. Besides this, he was responsible for keeping me busy with
mathematical problems, helping me out with clever ideas when I got stuck,
and for providing a relaxed work environment on floor MA 6-2.
I would also like to thank Gil Kalai, who agreed to be co-referee for this
thesis on rather short notice.
Peter McMullen shared his ideas on unneighborly polytopes and carefully read and responded to mine. He also copied large parts of Lockeberg’s
thesis for me and sent them to Berlin. Thanks!
Without Christos A. Athanasiadis, Chapters 3 and 4 would not exist.
Apart from this, I thank him for coming to Berlin to present his results on
viii
connectivity of incidence graphs of polytopes in the MDS lecture on June
16, 2008.
Micha A. Perles shared his knowledge on Perles’ Skeleton Theorem during a brief conversation at the “11th Midrasha Mathematicae” in Jerusalem.
Many of the figures in this thesis were programmed in PostScript. I
learned to program PostScript from Bill Casselmann in a course offered by
the BMS. Besides being the math graphics guru, he is also a really nice guy.
I received a lot of help from everyone at the BMS office: Nadja Wisniewski, Anja Bewersdorff, Tanja Fagel, and Mariusz Szmerlo.
Julian Pfeifle and his wonderful family, Lourdes, Nina, and Theo, made
sure that my family and I had a great time in Barcelona.
My friends in the Discrete Geometry Group made my time as a PhD student all the more enjoyable. In a very particular order (beauty? height? intelligence? age?), they are Axel Werner, Thilo Rörig, Raman Sanyal, Nikolaus Witte, Anna Gundert, Anton Dochtermann, Bruno Benedetti, Carsten
Schultz, and Benjamin Matschke. Out of these, Anna, Raman, Anton and
Axel deserve a special mentioning: They read parts of this thesis and helped
to improve the exposition significantly.
Elke Pose was of great help with administrative things. She also made
sure that the supply of mineral water did not run out.
I had a lot of fun presenting a program on Mathematik im Film together
with Thomas Vogt on two occasions during the Jahr der Mathematik (2008).
I have written this thesis in LATEX using the excellent memoir class by
Peter Wilson. Figures were either programed in PostScript, as mentioned,
or drawn in xfig.
Life is not all mathematics (or is it?), and some of the things that helped
me stay alive during the last years were sports (running in Kleistpark,
swimming in Stadtbad Schöneberg, and playing tennis with Till Plumbaum,
Axel Werner, Andreas Profous, and Sebastian Stiller ), learning Spanish
(at the Sprach- und Kulturbörse der TU Berlin in the highly enjoyable
courses by Florencia, Carmen, Liliana, and Miguel ), the brilliant pieces by
Les Luthiers, un grupo de música y humor de Argentina, que he conocido
gracias a Florencia, and some of the most interesting cities in the world:
San Francisco, Jerusalem, Barcelona, Paris, Amsterdam, Berlin. I also wish
to thank my parents for their continuing support.
The two most important people in my life deserve the final spot: Nora
and my wonderful epsilon Finja.
Berlin, December 2008
Ronald Frank Wotzlaw
ix
Contents
Preface
vii
Contents
xi
0 Introduction
1 The
1.1
1.2
1.3
1
Basics
Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Posets and Lattices . . . . . . . . . . . . . . . . . . . . . . .
Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I Connectivity of Polytope Skeleta
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7
10
13
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2 Refinement Homeomorphisms of Polytopes
23
2.1 Simplex Refinements of Polytopes . . . . . . . . . . . . . . . 27
2.2 Lockeberg’s Counterexamples . . . . . . . . . . . . . . . . . 30
2.3 Refinements of Centrally Symmetric Polytopes . . . . . . . . 33
3 Incidence Graphs of Polytopes
3.1 Balinski’s Theorem for Polytopes
3.2 Higher Incidence Graphs . . . . .
3.3 Connectivity of Incidence Graphs
3.4 Polytopality of Incidence Graphs
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4 Athanasiadis’ Conjecture on Incidence Graphs
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4.1 Regular Cell Complexes . . . . . . . . . . . . . . . . . . . . 54
xi
Contents
4.2
4.3
4.4
4.5
4.6
Athanasiadis’ Conjecture . . . . . . . . .
Graph Manifolds . . . . . . . . . . . . .
Basic Properties of Graph Manifolds . .
Connectivity of Graph Manifold Skeleta
Proof of Athanasiadis’ Conjecture . . . .
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54
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5 Linkages in Polytope Graphs
5.1 Linkages . . . . . . . . . . . . . . . . . .
5.2 Simplicial Polytopes and 3-Polytopes . .
5.3 Minimal Linkedness of Polytopes . . . .
5.4 Linkages in Polytopes with Few Vertices
5.5 Minimal Linkedness in Small Dimensions
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II
Perles’ Skeleton Theorem
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6 Skeleta of Polytopes with Few Vertices
101
6.1 Reconstruction of Skeleta . . . . . . . . . . . . . . . . . . . . 103
6.2 Perles’ Skeleton Theorem . . . . . . . . . . . . . . . . . . . . 105
7 Perles’ Skeleton Theorem for Polytopes
7.1 Empty Faces in Polytopes . . . . . . . . . . .
7.2 A Bound on the Number of Flat Empty Faces
7.3 Disjoint Empty Faces . . . . . . . . . . . . . .
7.4 Pyramidally Inequivalent Complexes . . . . .
7.5 Empty Simplices in Simplicial Polytopes . . .
8 Generalizations of Perles’ Skeleton Theorem
8.1 Strong PL Spheres . . . . . . . . . . . . . . .
8.2 Empty Faces in Strong PL Spheres . . . . . .
8.3 Simplex Refinements of PL Spheres . . . . . .
8.4 Perles’ Skeleton Theorem for PL Spheres . . .
8.5 Pyramidally Perfect Lattices . . . . . . . . . .
8.6 Boolean Intervals . . . . . . . . . . . . . . . .
8.7 Reconstruction of Skeleta . . . . . . . . . . . .
8.8 Relatively Complemented Lattices . . . . . . .
8.9 Empty Pyramids in Lattices . . . . . . . . . .
8.10 Empty Pyramids in Upper Intervals . . . . . .
8.11 Proof for Relatively Complemented Lattices .
8.12 Proof for Pyramidally Perfect Lattices . . . .
xii
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109
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149
Contents
III Unneighborly Polytopes
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9 Nonsimplicial Mani Polytopes
155
9.1 Illuminated Polytopes . . . . . . . . . . . . . . . . . . . . . 157
9.2 Mani’s Simplicial Illuminated Polytopes . . . . . . . . . . . 160
9.3 Nonsimplicial Mani Polytopes . . . . . . . . . . . . . . . . . 163
10 Counterexamples to Marcus’ Conjecture
10.1 Unneighborly Polytopes . . . . . . . . . . . . . . . . .
10.2 The Sizes of Minimal Positive Spanning Configurations
10.3 Counterexamples to Marcus’ Conjecture . . . . . . . .
10.4 Upper Bound on the Size . . . . . . . . . . . . . . . . .
11 Positive Spanning Sets in Oriented Matroids
11.1 Oriented Matroids . . . . . . . . . . . . . . . . . .
11.2 Oriented Matroids, Polytopes, and Duality . . . . .
11.3 Positive Spanning Sets in Oriented Matroids . . . .
11.4 The Topological Representation Theorem . . . . . .
11.5 Upper Bound on Minimal Positive k-Spanning Sets
Appendix
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169
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191
A Poset of Structures
191
Bibliography
193
List of Symbols
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Index
209
xiii
Chapter 0
Introduction
This thesis revolves around two topics in the theory of polytopes: incidence
graphs and missing faces of polytopes.
In this introduction we will take a “brief tour” through this thesis and
highlight some of the main results. Figure 0.1 shows the relations between
the different chapters, and we will explain the dependences in the following.
A good start into the topic are two articles by Kalai [62] [63].
The article Polytope skeletons and paths [63] from 1997 (updated 2004)
is a survey article on topics such as Balinski’s theorem on the connectivity of
polytope graphs and the Hirsch conjecture. Part I of this thesis is concerned
with connectivity questions of polytope skeleta and contains to a large part
results that are related to Kalai’s survey [63].
Balinski’s theorem states that graphs of d-polytopes are d-connected [5].
It has spawned many generalizations, questions, and conjectures. In the
work related to Balinski’s theorem we can identify three distinct directions
for generalizations. Authors have strived for
(a) graph properties that generalize d-connectedness, for example, d-rigidity
of polytope graphs [59] [63], subdivisions of complete graphs with two
prescribed “principal vertices” [46], or monotone versions of Balinski’s
theorem [54];
(b) structures that generalize polytopes, most notably the graph manifolds
by Barnette [8];
(c) larger classes of graphs associated to polytopes including the incidence
graphs introduced by Sallee [98] and Athanasiadis [3].
1
0. Introduction
Chapter 11
Chapter 4
Chapter 5
Chapter 8
Chapter 10
Chapter 7
Chapter 9
Chapter 6
Chapter 3
Chapter 2
Figure 0.1: Interdependence of chapters. A dashed line means a thematical
continuation. A solid line means a strong dependence on results, that is, a main
result of the earlier chapter is needed in the later chapter. A curved line means
a mild dependence on results. For example, some lemma might be reused in the
later chapter.
We will go in all three directions, further in some than in others.
Chapter 2 lays the foundation for later chapters. The most important
result that we prove in this chapter is a variant of Grünbaum’s theorem that
every polytope is a refinement of the simplex of the same dimension [50].
We apply this theorem in Chapters 3, 5, and 7.
In Chapter 2 we also introduce a number of “connectivity type” problems and conjectures that would follow from a strengthening of Grünbaum’s
theorem. The aforementioned problem on complete graph subdivisions by
Gallivan, Lockeberg & McMullen [46] is one of them. We present counterexamples, originally due to Lockeberg [69], to this strengthening.
A short side note on refinement homeomorphisms of centrally symmetric
polytopes concludes this chapter.
Incidence graphs of polytopes are considered under two different aspects
in Chapter 3. We define the (k, ℓ)-incidence graph as the graph on the set
of k-faces in which two faces are connected by an edge if they lie on a
common ℓ-face. These graphs were considered previously by Sallee [98] and
Athanasiadis [3].
We determine the connectivity of (k, ℓ)-incidence graphs for ℓ ≥ 2k + 1.
2
Balinski’s theorem serves as an important ingredient in the proof of this
result, and we establish a strong geometric version of d-connectedness of
polytopes.
Polytopality of incidence graphs is also discussed briefly in this chapter.
Athanasiadis [4] had asked for (k, ℓ)-incidence graphs that are also graphs
(that is, (0, 1)-incidence graphs) of other polytopes and we provide some
interesting examples.
A different question by Athanasiadis will keep us occupied throughout
Chapter 4. He determined the connectivity of (k, k + 1)-incidence graphs
of polytopes [3] and, motivated by a result by Fløystad [41], conjectured
that his result extended to Cohen-Macaulay regular cell complexes with
intersection property [3, Conjecture 6.2]. This conjecture is established in
this chapter by considering the question stripped off its algebraic ballast in
the context of Barnette’s graph manifolds.
The last chapter of Part I deals with linkages in polytope graphs. Linkages are an important concept in graph theory related to the work of Robertson & Seymour [95] on graph minor theory. In polytope graphs they were
first considered by Larman & Mani [66].
We prove some new results on linkages in Chapter 5, which is based on
joint work with Axel Werner. The main results of this chapter were published in [116]. I believe that Chapter 5 is a complete account of everything
that is known to date about linkages in polytopes. In particular, linkages
in polytopes with “few vertices” are discussed in great detail.
Some aspects of the analysis in Chapter 5 hint towards a theorem by
Perles on skeleta of polytopes with few vertices [89], and they lead us gently
into Part II of this thesis. This theorem by Perles, which we call from
now on Perles’ Skeleton Theorem (following Kalai [62]), appears in the
other article by Kalai that I had mentioned above, Some aspects of the
combinatorial theory of convex polytopes [62] from 1994. This article served
as an important inspiration for Part II of this thesis.
Perles’ Skeleton Theorem states that the number of combinatorial types
of k-skeleta of polytopes on d + γ + 1 vertices is bounded by a function of
k and γ that is independent of d.
With little knowledge of polytopes with few vertices one can easily prove
Perles’ Skeleton Theorem for γ = 1, that is, for d-polytopes on d+2 vertices.
The characterization of these polytopes [51, Theorem 6.1.4], which is for
example implied by a theorem we prove in Chapter 5, says that every such
polytope has the combinatorial type of an iterated pyramid over a direct
sum of two simplices. The k-skeleta of these polytopes are easily listed. For
example, there are only three types of graphs (1-skeleta) that can appear,
the complete graph on d + 2 vertices, the complete graph minus one edge,
3
0. Introduction
and the complete graph minus two disjoint edges. In contrast, there are
⌊d2 /4⌋ combinatorial types of d-polytopes on d + 2 vertices.
Chapter 6 is a brief introduction to topics related to Perles’ Skeleton
Theorem. In particular, the problem of bounding the number of combinatorial types is reduced to the problem of bounding the number of empty
pyramids of bounded dimension.
In Chapters 7 and 8 we give in total four proofs of Perles’ Skeleton
Theorem. The mostly geometric proof given in Chapter 7 is generalized to
strong PL spheres (that is, cellular PL spheres with intersection property)
in Chapter 8.
Chapter 8 also contains proofs of this result for two types of lattices that
both generalize face lattices of polytopes: the class of graded relatively
complemented lattices, and even more general, the class of pyramidally
perfect lattices.
Although the proofs given in Chapters 7 and 8 are distinct enough to
really consider them as different, they also share some characteristics. Three
out of the four proofs can be roughly divided into a local and a global part.
The local part consists mainly in understanding the behaviour of empty
pyramids in quotients.
The global part, which in all three cases is an induction by taking vertex
figures or upper intervals, puts together this local information in a careful
way. We have to ensure that we count all the missing faces, but that we
do not count too many too often (remember that we want to bound the
number of empty pyramids by a function that does not depend on d).
There are different tools that may be used for the global part, among
them:
(a) Grünbaum’s theorem on simplex refinements [50], which we also generalize to strong PL spheres,
(b) the Erdős-Rado Sunflower Lemma [40], as was done by Kalai [62], and
(c) an innocent looking lemma on “large simplex faces” of polytopes with
few vertices, see Lemma 5.4.2, that generalizes to “large boolean intervals” in pyramidally perfect lattices.
These tools yield the different bounds on the size of the set of vertices
in empty pyramids, which we call the kernel, as summarized in Figure 0.2.
The proof for pyramidally perfect lattices is slightly different, as the
technique of induction by taking quotients does not work here. Nevertheless,
the “large boolean intervals” play an important role here as well.
In Chapter 7 we also consider the related problem of bounding the number of disjoint empty faces. In this case we even get a characterization of
the extremal examples by applying a theorem that we already proved and
4
structure
technique
bound on kernel size
polytopes
refinement
(k + γ − 1)γ(γ + 1)k−1
sunflower
(k + γ − 1)! (2γ)k+γ−1
strong PL sphere
refinement
(k + γ − 1)γ(γ + 1)k−1
relatively complemented
boolean interval (k + γ − 1)(2γ)k
lattices
. . .2
pyramidally perfect latboolean interval |22{z }
tices
2k+γ−1
Figure 0.2: Different proof methods for Perles’ Skeleton Theorem and the resulting bounds on the kernel size. Here, k denotes rank.
used in a different context in Chapter 5. We also give a complete list of
graphs of d-polytopes on d + 3 vertices.
We study unneighborly polytopes in Chapters 9, 10, and 11. These
chapters make up Part III of this thesis.
A problem by Mani on illuminated polytopes [70], which are a special
case of unneighborly polytopes, is solved in Chapter 9.
Hadwiger had conjectured that an illuminated d-polytope, that is, a dpolytope in which every vertex lies on an inner diagonal, has at least 2d
vertices. This was disproved by Mani [70].
Mani [70] also gave a construction for simplicial illuminated polytopes
on the minimum number of vertices possible and asked for nonsimplicial
examples. We provide these examples and prove for which dimensions only
simplicial ones exist.
Chapter 10 deals with a conjecture by Marcus on the size of minimal
positive k-spanning vector configurations.
This conjecture can be interpreted via Gale duality in terms of unneighborly polytopes. It is disproved in the special case k = 2 by Mani’s
illuminated polytopes on the minimum number of vertices.
We construct from these examples counterexamples to Marcus’ conjecture for all k ≥ 2. We also identify special classes of unneighborly polytopes
that satisfy the conjectured bound, and we prove a general bound on the
size of minimal positive k-spanning configurations by using Perles’ Skeleton
Theorem.
The problem that we consider in Chapter 11 is a similar question on the
size of minimal positive k-spanning sets in oriented matroids. It was posed
by Bienia & Las Vergnas [23, Exercise 9.35(iv)*].
5
0. Introduction
Once again an application of Perles’ Skeleton Theorem yields a solution
to this problem. We achieve the best bound on the size with the version
for strong PL spheres via the Topological Representation Theorem for oriented matroids by Folkman & Lawrence [42], Edmonds & Mandel [39], and
Lawrence [68].
The appendix contains a poset that relates (by inclusion) the different structures and combinatorial abstractions of geometric objects that are
relevant to this thesis.
Questions, open problems, and conjectures are scattered throughout the
text.
6
Chapter 1
The Basics
This chapter collects general notations and notions used throughout this
thesis. Special terminology, or terminology that pertains to only a single
chapter, is defined when it first appears.
The material on graph theory is based on the third edition of Diestel’s
graph theory book [38]. The part on posets and lattices is taken from Stanley’s book [107, Chapter 3] and partly from Ziegler’s book on polytopes [118,
Definition 2.5]. The basics in polytope theory are drawn from the books by
Ziegler [118] and Grünbaum [51]. The reader acquainted with these sources
should have no difficulty following later chapters without having read this
one.
We write R for the real numbers, N for the natural numbers, and we
use the notation [n] := {1, . . . , n}.
1.1
Graphs
A graph G¡is ¢a pair (V, E) consisting of a finite set V , the vertices of G, and
a set E ⊆ V2 , the edges of G. If G is a graph with vertices V and edges E,
we write V (G) := V and E(G) := E. We write uv := {u, v} for the edge
{u, v}. The vertices u and v are called the endpoints of the edge uv.
Two graphs G1 and G2 are called isomorphic if there is a bijection
φ : V (G1 ) → V (G2 ) such that uv ∈ E(G2 ) if and only if φ(u)φ(v) ∈ E(G2 ).
In this thesis, we will usually not distinguish between isomorphic graphs.
In particular, we will write Kn for any graph that is isomorphic to the
7
1. The Basics
(a) A graph.
(b) A subgraph that is a cy- (c) The subgraph induced
cle, but not induced.
by the vertices of the cycle.
Figure 1.1: Subgraphs and induced subgraphs.
¡ ¢
) on n vertices.
complete graph G = ([n], [n]
2
A graph is called bipartite if it admits a partition of its vertex set into
two classes such that every edge has its endpoints in different classes.
We write Km,n for the complete bipartite graph. This is the bipartite
· 2 with |V1 | = m, |V2 | = n in which all edges
graph on vertex set V = V1 ∪V
between V1 and V2 exist.
1.1.1
Subgraphs
If G1 and G2 are graphs with V (G1 ) ⊆ V (G2 ) and E(G1 ) ⊆ E(G2 ), then
G1 is a subgraph of G2 . If E(G1 ) = {uv ∈ E(G2 ) : u, v ∈ V (G1 )}, then the
graph G1 is an induced subgraph of G2 .
Let G = (V, E) be a graph and v ∈ V . We write N (v) for the neighbors
of v, that is, N (v) := {u ∈ V : uv ∈ E}.
A path of length k is a nonempty graph P = (V, E) of the form
V = {v0 , v1 , . . . , vk },
E = {v0 v1 , v1 v2 , v2 v3 , . . . , vk−1 vk },
where the vi are all distinct. We say that the path P connects or joins
v0 and vk . The vertices v1 , . . . , vk−1 are the inner vertices of P . We
will often notationally orient the path P by writing P as v0 v1 . . . vk−1 vk
or (v0 , e1 , v1 , . . . , vk−1 , ek , vk ), where ei = vi−1 vi for every i = 1, . . . , k. If P1
and P2 are paths such that their union is a path as well, then we call this
union the concatenation of P1 and P2 and denote it by P1 P2 . A subpath of a
path P is a subgraph of P that is a path. Distinct paths P1 , P2 , . . . , Pm are
independent if none of them contains an inner vertex of one of the others.
A walk of length k is a sequence (v0 , e1 , v1 , . . . , vk−1 , ek , vk of vertices
such that ek = vi vi+1 is an edge for every i = 0, . . . , k − 1. Clearly, a walk
contains a path that connects v0 to vk .
8
1.1. Graphs
(a) The join operation applied to a cycle
of length 5 and a path of length 3.
(b) The join operation in the complement.
Figure 1.2: The join of two graphs.
A cycle of length k is a graph of the form
V = {v0 , v1 , . . . , vk },
E = {v0 v1 , v1 v2 , v2 v3 , . . . , vk−1 vk , vk v0 },
where the vi are all distinct.
Figure 1.1 shows a graph, a cycle of length 4 in that graph, and the
subgraph induced by the vertices of the cycle.
1.1.2
Connectivity
A graph G is connected if for every two vertices u, v ∈ V (G) there is a path
in G that connects u and v. A graph G is k-connected if |V (G)| ≥ k + 1
and the graph G \ U is connected for every set U ⊆ V (G) of size less
than k. The greatest integer k such that the graph G is k-connected is the
connectivity κ(G).
We need the following important result by Menger [80]; see also [38,
Theorem 3.3.6].
Theorem 1.1.1 (Menger’s theorem [80]). A graph G is k-connected if
and only if for every two vertices u and v there are k distinct independent
paths connecting u and v.
1.1.3
Graph constructions
If G1 and G2 are graphs with disjoint vertex sets we write G1 ∗ G2 for the
join of G1 and G2 defined by the vertex set V (G ∗ H) := V (G) ∪ V (H) and
the set of edges
E(G ∗ H) := E(G) ∪ E(H) ∪ {vv ′ : v ∈ V (G), v ′ ∈ V (H)}.
9
1. The Basics
v
u
(a) A 3-connected graph with three independent paths drawn between the vertices
u and v.
(b) The complete graph on 4 vertices is a
topological minor of the graph shown. A
subdivision of K4 is drawn bold.
Figure 1.3: Connectivity and topological minors.
The complement graph G of a graph G is the graph on vertex set V (G)
in which uv is an edge if and only uv is not an edge of G.
The join operation is exemplified in Figure 1.2(a). The displayed graph
is the join of a cycle of length 5 and a path of length 3. In most cases it
is more transparent to display the result in the complement graph. The
complement of the join is the disjoint union of the complements; see Figure 1.2(b).
1.1.4
Minors
A graph G is a minor of a graph H if G is obtained from H by deleting
vertices and contracting edges.
A graph G′ is a subdivision of G (or a refinement) if G′ is obtained from
G by replacing edges by pairwise independent paths (possibly of length one).
If G′ is a subgraph of another graph H then G is said to be a topological
minor of H. A topological minor is always also a minor, but not conversely.
1.2
Posets and Lattices
A finite partially ordered set (or short poset) is a pair (L, ≤) consisting of
a finite set L and a binary relation ≤, the partial order on L, that satisfies
the following three axioms:
(i) For all x ∈ L we have x ≤ x (reflexivity).
(ii) If x ≤ y and y ≤ x, then x = y (antisymmetry).
(iii) If x ≤ y and y ≤ z, then x ≤ z (transitivity).
10
1.2. Posets and Lattices
We simply write L for the poset (L, ≤).
Let L1 and L2 be partially ordered sets. Then L1 and L2 are isomorphic,
in notation L1 ∼
= L2 , if there is an order-preserving bijective map π : L1 →
L2 whose inverse is again order-preserving, that is, x ≤ y in L1 if and only
if π(x) ≤ π(y) in L2 .
If L is a poset, we denote by Lop the opposite poset. This is the poset
on the set L such that x ≤ y in Lop if y ≤ x in L for all x, y ∈ L.
1.2.1
Intervals, Chains, and Rank
For x, y ∈ L with x ≤ y we write
[x, y] := {z ∈ L : x ≤ z ≤ y}
and call [x, y] the (closed) interval between x and y.
An element y is said to cover an element x, denoted by x ⋖ y, if x < y
and [x, y] = {x, y}.
The poset L has a 0̂ if there is an element 0̂ ∈ L with 0̂ ≤ x for all
x ∈ L. It has a 1̂ if there is an element 1̂ ∈ L with x ≤ 1̂ for all x ∈ L.
Intervals of type [0̂, x] with x ∈ L are called lower intervals, intervals of
type [x, 1̂] are called upper intervals.
A chain in a poset L is a totally ordered subset X ⊆ L. The length of
a chain X is |X| − 1. The poset L is graded if every maximal chain has the
same length. If L is graded, the rank rk(x) of x ∈ L is defined to be the
maximum length of a chain in the interval [0̂, x]. The rank of L, denoted
by rk(L), is the maximum length of a chain in L.
1.2.2
Lattices
If every two elements x and y of a poset L have a unique upper bound
x ∨ y, the join of x and y, we say that L is a join-semilattice. If every two
elements x and y have a unique lower bound x ∧ y, the meet of x and y, we
say that L is a meet-semilattice.
A lattice is a finite partially ordered set that is both a join-semilattice
as well as a meet-semilattice. The following is a useful criterion for telling
when a poset is a lattice.
Proposition 1.2.1 (see [107, Proposition 3.3.1]). Let L be a finite
meet-semilattice with 1̂. Then L is a lattice. Dually, a join-semilattice with
0̂ is a lattice.
11
1. The Basics
1.2.3
Atoms and Coatoms
If L is a meet-semilattice, which has a 0̂, then the atoms A(L) of L are the
elements that cover 0̂. If L is a join-semilattice, which has a 1̂, then the
coatoms of L are the elements that are covered by 1̂.
A lattice L is atomic if every element x 6= 0̂ is the join of k ≥ 1
atoms, that is, there are a1 , . . . , ak ∈ A(L) with x = a1 ∨ a2 ∨ · · · ∨ ak .
It is coatomic if every element x 6= 1̂ is the meet of k ≥ 1 coatoms.
We write A(x) for the atoms below element x.
1.2.4
Boolean lattices
Let A = {a1 , . . . , an } be a set of n elements. We denote by B(A) or
B(a1 , . . . , an ) the boolean lattice of rank n on atoms a1 , . . . , an , that is,
B(a1 , . . . , an ) := B(A) := (2A , ⊆).
We say that L is boolean if it is isomorphic to a boolean lattice.
1.2.5
Relatively complemented lattices
A lattice satisfies the diamond property if every interval of length 2 is
boolean, that is, if it has exactly 4 elements.
A lattice L is complemented if for each element x there is an element y
such that x ∨ y = 1̂ and x ∧ y = 0̂. It is relatively complemented if every
interval [x, y] ⊆ L is complemented.
The following theorem gives a useful criterion for deciding when a given
lattice is relatively complemented. It implies that every lattices that satisfies the diamond property is also relatively complemented.
Theorem 1.2.2 (Björner [20]). A finite lattice is relatively complemented
if and only if every interval of length 2 has cardinality at least 4. Every
relatively complemented lattice is both atomic and coatomic.
There are indeed lattices that do not satisfy the diamond property, but
neverthelesss are relatively complemented. One such lattice is shown in
Figure 1.4.
12
1.3. Polytopes
(a) A boolean interval.
(b) An interval of length 2 that is not
boolean.
Figure 1.4: A lattice that does not satisfy the diamond property, but that is
relatively complemented.
1.3
Polytopes
Let V be a subset of Rd . We denote by
k
k
X
X
aff(V ) := {
λi vi : {v1 , . . . , vk } ⊆ V,
λi = 1},
i=1
k
X
cone(V ) := {
i=1
k
X
conv(V ) := {
i=1
i=1
λi vi : {v1 , . . . , vk } ⊆ V, λi ≥ 0},
λi vi : {v1 , . . . , vk } ⊆ V,
k
X
i=1
λi = 1, λi ≥ 0}
the affine hull, conical hull or cone, and convex hull, respectively, generated
by the set V .
An affine k-dimensional subspace of Rd , which can always be written as
the affine hull of k + 1 affinely independent points, will be called a k-flat.
A k-flat A and an ℓ-flat B are skew if dim(aff(A ∪ B)) = k + ℓ + 1.
A polytope P is the convex hull of a finite set of points of Rd . Thus if
V = {v1 , . . . , vn } is a set of n points, we can write a polytope as
n
n
X
X
P = conv(V ) = {
λi v i :
λi = 1, λi ≥ 0}.
i=1
i=1
Equivalently, we can write every polytope as the bounded intersection of a
finite number of halfspaces—this is the Main theorem for polytopes; see [118,
Theorem 1.1].
Using this equivalence, one easily shows that the intersection of a polytope with an affine space is a polytope and that the projection of a polytope
is again a polytope.
13
1. The Basics
The dimension dim(P ) of a polytope P is the dimension of its affine
hull, and we say that P is a dim(P )-polytope. If d = dim(P ), the polytope
P is full-dimensional.
A face of a polytope is a set of the form
F = P ∩ {x ∈ Rd : aT x = b},
where aT x ≤ b is a valid inequality for P . With this definition, the polytope P itself and the empty set ∅ are faces—they are given by the valid
inequalities 0T x ≤ 0 and 0T x ≤ 1, respectively (the symbol 0 denotes the
column vector in which every entry is zero). A proper face is a face that
is not P , and ∅ is the only trivial face. Clearly, a face of a polytope is a
polytope. The faces of dimension 0, 1, dim(P ) − 2, and dim(P ) − 1 are
called vertices, edges, ridges, and facets, respectively. We write V(P ) for
the set of vertices of P .
If P is a d-polytope in Rd , then int P is the set of interior points of P ,
that is, the set of all points that do not lie on a face of dimension smaller
than d. If P is a k-polytope that is embedded in a k-flat H of Rd , then
the relative interior points of P are the interior points of P considered as a
full-dimensional polytope in H ∼
= Rk .
A flag in a polytope is a sequence of faces such that every face is properly
contained in the next face of the sequence.
The set of faces of P ordered by inclusion forms a poset L(P ), called
the face poset of P .
The standard d-simplex is the polytope
∆d := conv{e1 , e2 , . . . , ed+1 } ⊂ Rd+1 .
This is a d-dimensional polytope.
The standard d-cube is the polytope
Cd := conv{{+1, −1}d } ⊂ Rd .
We also denote the 2-cube by ¤. This polytope is also called a quadrilateral.
The standard d-crosspolytope is the polytope
Cd∆ := conv{±e1 , ±e2 , . . . , ±ed } ⊂ Rd .
In dimension d = 3, we will also call C3∆ the octahedron.
1.3.1
Vertex Figures and Face Figures
Let P be a d-polytope in Rd , where d is at least one. Let v be a vertex
of P , and take an inequality aT x ≤ b0 that defines this vertex, so that we
14
1.3. Polytopes
have {v} = P ∩ {x ∈ Rd : aT x = b0 }. If b1 < b0 is a real number such that
aT p < b1 for all p ∈ V(P ) \ {v}, we call the polytope
P/v := P ∩ {x ∈ Rd : aT x = b1 }
a vertex figure of P at v. This is a full-dimensional polytope in the (d − 1)dimensional space {x ∈ Rd : aT x = b1 } ∼
= Rd−1 .
This construction generalizes as follows: Let F be a face of P of dimension k ≥ 0. If dim F = 0, then P/F is the vertex figure at F . Otherwise,
choose a vertex v of F and inductively define P/F := (P/(F/v))/v. We call
P/F a face figure of P at F . The polytope P/F is a polytope of dimension
d −k −1. It can be obtained from P by a cut with a suitable (d −k −1)-flat.
1.3.2
Polarity
Every polytope P has a polytope that has an anti-isomorphic poset. If we
assume that P contains the origin in its interior and is given by
P = {x ∈ Rd : Ax ≤ b},
where A is a real m × d matrix and b ∈ Rm , then the polar of P is the
polytope
P ∆ = {y ∈ Rd : y = cT A, c ≥ 0, cT 1 = 1};
see [118, Section 2.3].
The face poset of P ∆ is anti-isomorphic to the poset of P . This gives
a new interpretation for face figures: Taking a face figure of P corresponds
to taking a face of P ∆ .
We write F 3 for the face of P ∆ that corresponds to the face F of P .
1.3.3
Properties of the Face Lattice
The face poset of a polytope has a number of specific properties that we
list in the following theorem.
Theorem 1.3.1 (see [118, Theorem 2.7]). Let P be a d-polytope.
(i) The face poset L(P ) is a lattice. We will therefore also speak of it as
the face lattice of P .
(ii) The lattice L(P ) is graded with rank function rk(F ) := dim(F ) + 1.
(iii) The lattice L(P ) is atomic and coatomic.
15
1. The Basics
(iv) The lattice L(P ) has the diamond property, that is, every interval of
length 2 has exactly 4 elements.
(v) The opposite lattice L(P )op is the face lattice of the polar polytope P ∆ .
(vi) Every interval [F, G] is the face lattice of a (dim(G) − dim(F ) − 1)polytope, namely the face lattice of G/F .
1.3.4
Combinatorial Types
Two polytopes P1 and P2 are combinatorially isomorphic if their face lattices L(P1 ) and L(P2 ) are isomorphic.
All the following definitions are clearly invariant under combinatorial
isomorphisms.
Every polytope that is combinatorially isomorphic to ∆d , Cd , or Cd∆ is
called a d-simplex, d-cube, or d-crosspolytope, respectively.
A polytope P is simplicial if all its facets are simplices, that is, every
facet contains exactly dim(P ) many vertices. It is simple, if every vertex
is contained in exactly d facets. A polytope is simplicial if and only if its
polar polytope is simple.
A polytope is ℓ-simplicial, if all faces up to dimension ℓ are simplices.
(The polar notion of ℓ-simplicity does not appear in this thesis.)
We define a vector f (P ) = (f−1 , f0 , . . . , fd−1 ) with integral entries by
letting fk be the number of k-faces of P . Thus f−1 = 1, f0 is the number of
vertices, f1 is the number of edges, fd−2 is the number of ridges, and fd−1
is the number of facets. The vector f (P ) is the f -vector of the polytope P .
We write fk (P ) for the kth entry of f (P ).
A related notion is the h-vector. Given the f -vector of a simplicial
polytope, the h-vector is defined componentwise by
hk :=
k
X
k−i
(−1)
i=0
µ
¶
d−i
fi−1 ,
d−k
for every k ∈ {0, . . . , d}.
Finally, we define the g-vector for simplicial polytopes by g0 := h0 = 1,
and gk := hk − hk−1 for every k ∈ {1, . . . , ⌊d/2⌋}.
The definition of the h- and g-vectors for general polytopes is much
more delicate; see for example [62, Section 4] and [13]. As we will we not be
dealing with higher entries of the g-vector, we define γ(P ) := f0 − d − 1 =
g1 (P ).
16
1.3. Polytopes
1.3.5
Gale Duality
A Gale diagram of a polytope on d + γ + 1 vertices is a vector configuration,
that is, a finite multiset of vectors, in Rγ that encodes the combinatorial
data of the polytope.
For detailed information on Gale duality, in particular, how they are
constructed, we refer the reader to the extensive survey by McMullen [78],
the exposition in [118] together with a crash course in oriented matroid
theory, or the elementary account in [74].
Gale duality of polytopes is intimately related to the notion of positively
spanning.
Definition 1.3.2 (Positively spanning). Let V = (v1 , . . . , vn ) be a vector
configuration in Rr . We call V positively spanning for a vector space W ⊆
Rr if
cone(V ) = {t1 v1 + · · · + tn vn : (t1 , . . . , tn ) ≥ 0} = W,
that is, nonnegative combinations of the vectors in V span the space W .
Here we state the fundamental principle for reading off the combinatorics
of a polytope from one of its Gale diagrams.
Theorem 1.3.3. Let P be a d-polytope on n = d + γ + 1 vertices v1 , . . . vn ,
and let A = (a1 , . . . , an ) be a Gale diagram of P , where the ai are vectors
in Rγ .
Then I ⊆ [n] is the index set of a face F of P , that is, the set of vertices
of F is exactly {vi : i ∈ I}, if and only if the set A′ = {aj : j ∈
/ I} is
positively spanning for its linear span.
Figure 1.5 displays a Gale diagram in R2 , where we indicate how many
times a vector appears in the configuration by a label next to it. We see
that the corresponding polytope has 4 · 2 = 8 vertices, and so its dimension
is d = 8−3 = 5. Theorem 1.3.3 implies that two vertices that correspond to
the two copies of a vector in Figure 1.5 do not form an edge of the polytope.
For d ≥ 0, γ ≥ 0 we define P γd to be the class of d-polytopes on d + γ + 1
vertices. Further define S γd to be the subclass of P γd of simplicial polytopes.
All the polytopes in P γd have γ-dimensional Gale diagrams.
1.3.6
Polytope Constructions
If P1 and P2 are two polytopes of dimensions d and e, then P1 × P2 denotes
the product of P1 and P2 , obtained by taking the cartesian product of P1
and P2 . This is a (d + e)-dimensional polytope. If P2 = I is a line segment,
17
1. The Basics
2
2
2
2
Figure 1.5: Gale diagram of a 5-polytope on 8 vertices with 4 disjoint missing
edges.
then the product P1 × I is also called the prism over P1 . Iterating the prism
with I = [−1, 1] starting from a point, we get a translation of the standard
d-cube.
By P1 ∗P2 we denote the join of two polytopes P1 and P2 . This operation
was apparently first introduced in [105]; see also [51, Exercise 4.8.1].
Geometrically, the join of polytopes P1 and P2 is obtained by placing
them in skew affine spaces of dimensions dim(P1 ) and dim(P2 ), respectively,
and taking the convex hull.
Combinatorially, the faces of P1 ∗ P2 are the joins of faces of P1 and P2 .
The dimension of P1 ∗ P2 is dim(P1 ) + dim(P2 ) + 1.
The join specializes to the operation of taking a pyramid over a polytope.
If P is a (d − 1)-polytope in a hyperplane of Rd , and a is a point outside of
that hyperplane, then pyra (P ) := (conv P ∪ {a}) = P ∗ {a} is the pyramid
over P with apex a and base P . We will also simply write pyr(P ) for
a pyramid over P , and we call a polytope that is a pyramid pyramidal.
Furthermore, we denote by pyrk (P ) the k-fold pyramid over P , that is, the
polytope obtained by iterating the pyramid operation k times. A (d+1)-fold
pyramid over the empty set is a d-simplex.
For faces F ⊆ P1 and G ⊆ P2 we denote by (P1 , F )⊕(P2 , G) the subdirect
sum of the polytopes P1 and P2 with respect to F and G [77].
This operation can be realized geometrically by placing P1 and P2 in
affine spaces that intersect exactly in one point x0 that is a relative interior
point of F and G.
The combinatorial description is given by the following proposition.
Proposition 1.3.4 (McMullen [77]). The faces of (P1 , F ) ⊕ (P2 , G) fall
18
1.3. Polytopes
into two categories. A face is either given by
(i) F1 ∗ G1 , where F1 ∩ F ⊂ F and G1 ∩ G ⊂ G, or
(ii) (F1 , F ) ⊕ (G1 , G), where F ⊆ F1 and G ⊆ G1 .
We call the special subdirect sum (P1 , v0 ) ⊕(P2 , P2 ), where v0 is a vertex
of P1 , a vertex sum. If P2 is a line segment, a vertex sum is called a vertex
split.
We write P1 ⊕ P2 := (P1 , P1 ) ⊕ (P2 , P2 ) and call this the direct sum of
the polytopes P1 and P2 ; see [51, Exercise 4.8.4].
Combinatorially, the boundary of P1 ⊕ P2 is the join of the boundaries
of P1 and P2 , that is, the proper faces of P1 ⊕ P2 are the joins of proper
faces of P1 and P2 . The dimension of P1 ⊕ P2 is dim(P1 ) + dim(P2 ).
A special case of the direct sum is the sum of a polytope P and an
interval I, which yields the bipyramid bipyr P = P ⊕ I. We call P a base
of the bipyramid. The two vertices of I are the apexes of the bipyramid.
Iterating the bipyramid with I = [−1, 1] starting from a point, we obtain a
translation of the standard d-crosspolytope.
1.3.7
Polytopal Complexes
Definition 1.3.5 (Polytopal complex). A polytopal complex C in Rd is
a finite collection of polytopes, called the faces of C, in Rd such that
(i) the empty set ∅ is in C,
(ii) if P ∈ C and F ⊆ P is a face of P , then F ∈ C, and
(iii) the intersection P ∩ Q of polytopes P, Q ∈ C is a face of both P and
Q, that is, the complex satisfies the intersection property.
The dimension of a polytopal complex is the maximum over the dimensions
of its faces.
We write V(C) for the set of vertices of C, that is, V(C) = ∪P ∈C V(P ).
Special cases of polytopal complexes are the complex of all faces of a
polytope and the boundary complex B(P ) of a polytope, that is, the complex
of all proper faces of a polytope.
Let C be a polytopal complex. The face poset F(C) of C is the set
of faces of C partially ordered by inclusion. Two polytopal complexes are
combinatorially isomorphic if their face posets are isomorphic.
19
1. The Basics
Figure 1.6: The vertex figure at the pulled vertex is isomorphic to the link.
A subcomplex of a polytopal complex is a subset of its faces that is
again a polytopal complex. A polytopal complex has certain distinguished
subcomplexes that we define in the following.
Let C be a polytopal complex in Rd . The k-skeleton skelk (C) is the
polytopal complex of faces of C of dimension at most k, that is,
skelk (C) := {F : F ∈ C, dim(F ) ≤ k}.
Other important subcomplexes are the star, antistar, and the link1 .
They are defined in the following definition.
Definition 1.3.6 (Star, antistar, link). Let C be a polytopal complex
and let F be a face of C. We define
• the star of F by
starC (F ) := {G ∈ C : there is H ∈ C such that F ⊆ H and G ⊆ H},
• the antistar of F by astC (F ) := {G ∈ C : F 6⊆ G},
• and the link at F by linkC (F ) := astC (F ) ∩ starC (F ).
The link of a vertex in the boundary complex of a d-polytope is a polytopal complex that is combinatorially isomorphic to the boundary complex
of a (d − 1)-polytope. It is isomorphic to the vertex figure after “pulling”
the vertex; see Figure 1.6.
If F ⊆ C is a set of faces of a polytopal complex, we denote by || F || =
∪ F ⊂ Rd the underlying space of F.
1
There seems to be no consensus on the meaning of the term link in the polytope
literature. For example, Billera & Björner define this term to mean what we call face
figure [16, p. 421]. We use it in the same way as Ziegler [118, p. 237].
20
Part I
Connectivity of Polytope Skeleta
Chapter 2
Refinement Homeomorphisms of Polytopes
A refinement homeomorphism between polytopes P and Q is a homeomorphism φ : P → Q that maps any face of P to a union of faces of Q; see also
Definition 2.1.1.
Refinement homeomorphisms of polytopes are useful for a number of
purposes. In this thesis, they are used to determine the connectivity of
incidence graphs of polytopes in Section 3.3, to prove a lower bound on the
linkedness of general polytopes in Section 5.3, and to prove Perles’ Skeleton
Theorem; see Chapters 7 and 8, in particular Sections 7.2 and 8.4.
The basic statement needed for these applications, Grünbaum’s theorem
that every d-polytope is a refinement of the d-simplex [50], is proved in
this chapter. Our version asserts that one may preassign a flag of faces
as principal faces of the refinement, as was proposed by Grünbaum [51,
Exercise 11.1.3], and certain technical conditions that we need in Chapter 3
and Chapter 7. (A principal face is a face such that the preimage in ∆d is
a face.)
Because of the usefulness of this theorem some authors have asked for
extensions. The following conjecture was raised for example in Grünbaum’s
book in a stronger version [51, Exercise 11.1.4], and by Jockusch & Prabhu;
see [56, Conjecture 3].
Conjecture 2.0.1 (disproved [69]). Let P be a d-polytope and v1 , v2 be
distinct vertices of P . Then there is a refinement homeomorphism from the
d-simplex to P , such that v1 and v2 are principal faces of the refinement.
23
2. Refinement Homeomorphisms of Polytopes
This conjecture, if it were true, would have implied solutions to a number
of problems or conjectures that we discuss in the following.
The first one is the restriction of Conjecture 2.0.1 to the 1-skeleton.
Problem 2.0.2 (Gallivan, Lockeberg, and McMullen [46]). Let P be
a d-polytope. Does G(P ) contain, for every x, y ∈ V (P ), a refinement of
Kd+1 in which x and y are both principal?
The answer to this question is affirmative in the special case of simplicial
polytopes. This was shown by Larman & Mani [66].
Gallivan, Lockeberg & McMullen [46] have shown that, in general, one
may not preassign three arbitrary vertices of P as principal vertices of a
Kd+1 - refinement. Precisely, they have proven that (a) if d = 3 one may
preassign three vertices as principal, but not four, and (b) for all d ≥ 4, there
is a d-polytope P with distinct vertices x, y, z that cannot all be principal
in any Kd+1 -refinement in G(P ).
The second problem is Lockeberg’s conjecture. To state it concisely we
need the following definition of strong chains.
Definition 2.0.3. Let P be a d-polytope and 0 ≤ k ≤ d. A strong k-chain
C in P is a union of k-faces C = F1 ∪ · · · ∪ Fm such that Fi ∩ Fi+1 is a
(k − 1)-face for all i ∈ {1, . . . , m − 1}.
Conjecture 2.0.4 (Lockeberg’s conjecture [69]). Let P be a d-polytope,
v1 , v2 ∈ V (P ), and let d = d1 +· · ·+dr for positive integers d1 , . . . , dr . Then
there are strong di -chains Ci , for i = 1, . . . , r, such that Ci ∩ Cj = {v1 , v2 },
for i 6= j.
This conjecture is also mentioned by McMullen [48, Problem 57], in a
survey by Kalai [63, Conjecture 19.5.4], and by Jockusch & Prabhu [56,
Conjecture 1].
Lockeberg’s conjecture is a strengthening of Balinski’s theorem: If we
set di = 1 for all i = 1, . . . , r = d, then the statement reduces to dconnectedness. Furthermore, a strong di -chain is di -connected. Thus Lockeberg’s conjecture implies that one may choose the d independent paths from
v1 to v2 on the boundaries of the di -chains, i = 1, . . . , r.
In contrast to this, it is not difficult to see that Lockeberg’s conjecture
is a weakening of Conjecture 2.0.1.
See Figure 2.1(a) for an example of strong chains.
The third problem whose solution would have followed from a positive
solution to Conjecture 2.0.1 is a problem by Prabhu [93]; see also Jockusch
& Prabhu [56, Conjecture 2], and Ziegler [118, Exercise 3.14]. For the
statement we need the following definition of k-paths.
24
v2
v2
v1
v1
(b) Two 2-paths between the vertices v1
and v2 . These paths are disjoint in the
sense of Definition 2.0.5 as they do not
share a 2-face.
(a) Two strong 2-chains and a strong 1chain that intersect in the vertices v1 and
v2 . Lockeberg’s conjecture asserts that we
can find a subcomplex similar to this one
in every 5-polytope.
Figure 2.1: Strong k-chains and disjoint k-paths.
Definition 2.0.5. Let P be a d-polytope, v1 , v2 ∈ V (P ), and 0 ≤ k ≤ d.
A k-path between v1 and v2 is a strong k-chain F1 ∪ F2 ∪ · · · ∪ Fm such that
v1 ∈ V (F1 ), v2 ∈ V (Fm ). Two k-paths are said to be disjoint if they have
no k-face in common.
See Figure 2.1(b) for two 2-paths between vertices v1 and v2 .
It is important to observe that the notion of disjointness of k-paths is
very different from the disjointness of strong chains in Lockeberg’s conjecture; see Figure 2.1. In that respect, the following problem is more in line
with the material in Chapters 3 and 4.
Question 2.0.6 (Prabhu [93]). Are there
two vertices v1 and v2 of any d-polytope?
¡d¢
k
disjoint k-paths between any
The answer is affirmative for the d-simplex. Thus we could infer the
general case indeed from Conjecture 2.0.1 by taking a refinement with principal vertices v1 and v2 . Jockusch & Prabhu [56] have shown that a natural
geometric approach to Question 2.0.6 fails.
25
In Section 2.2 we present counterexamples to Conjecture 2.0.1. We
thereby rule out this approach to the aforementioned problems. The examples are due to Lockeberg [69], who was a PhD student of McMullen.
There are two reasons why these examples are included here: First, the
explanation given here why no such refinement exists is simpler than the
one given by Lockeberg. Second, so far these examples have only appeared
in Lockeberg’s dissertation, which is not easily available (McMullen was so
kind to copy the relevant chapters for us. Thanks!). I believe there is only
one reference to them in the literature, see McMullen [78, 3B17], but even
there the examples themselves are not given.
We then consider refinements of centrally symmetric polytopes related
to the following conjecture by Grünbaum [51].
Conjecture 2.0.7 (Grünbaum [51, Exercise 11.1.5, p. 205]). It may
be conjectured that for each d there exists a finite family {Pi | 1 ≤ i ≤ n(d)}
of centrally symmetric d-polytopes such that for each centrally symmetric dpolytope P the complex B(P ) is a refinement of at least one of the complexes
B(Pi ), for 1 ≤ i ≤ n(d).
It was remarked in the second edition of Grünbaum’s book, see [51,
p. 224b], that this conjecture may be related to the “3d -conjecture” by
Kalai [61].
Exercise 11.1.5 may be related to the conjecture by Kalai that if a dpolytope is centrally symmetric, then it must have at least 3d non-empty
faces.
Kalai conjectures that equality is achieved only by the Hanner polytopes
that can be generated from an interval by taking products and dualization
(which includes the cubes and the cross polytopes). One may speculate
that this also provides the finite family needed for exercise 11.1.5.
We reformulate this remark as the following conjecture.
Conjecture 2.0.8 ((disproved)). Every centrally symmetric d-polytope
is a refinement of some d-dimensional Hanner polytope.
Clearly, if P is a refinement of a polytope Q, then the f -vector of P is
componentwise at least as large as the f -vector of Q. Recent examples by
Sanyal, Werner & Ziegler [99] imply that Conjecture 2.0.8 is false for all
d ≥ 4.
26
2.1. Simplex Refinements of Polytopes
2.1
Simplex Refinements of Polytopes
Definition 2.1.1 (Refinement homeomorphism, principal face). Let
K and C be polytopal complexes. A refinement homeomorphism is a homeomorphism φ : K → C such that the image of a face of K is a union of faces
of Q.
A refinement homeomorphism between polytopes is a homeomorphism
φ : P → Q such that the image of a face of P is a union of faces of Q.
A principal face of a refinement homeomorphism φ is a face F of Q such
that φ−1 (F ) is a face of P .
The following theorem is a strengthening of a theorem by Grünbaum [50];
see also [51, Section 11.1, p. 200] [51, Exercise 11.1.3].
Theorem 2.1.2. Let P be a d-polytope, and
∅ = F−1 ⊂ F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fd−1 ⊂ Fd = P
a complete flag of faces. Then there is a refinement homeomorphism φ :
∆d → P such that the following statements hold for every i ∈ {−1, 0, . . . , d−
1}:
(i) The face Fi is a principal face of the refinement.
(ii) The image of an ℓ-face of ∆d , with i ≤ ℓ ≤ d, that contains φ−1 (Fi ),
which is a face by (i), contains only ℓ-faces of P that contain Fi .
Furthermore, the principal vertices φ(V (∆)) are affinely independent.
Proof. The proof proceeds by induction on the dimension d. For d = 0 the
statement clearly holds.
Let d ≥ 1, take the vertex v := F0 and consider the vertex figure P/v.
By induction, the statement holds for P/v and the flag
∅ = F0 /v ⊂ F1 /v ⊂ F2 /v ⊂ · · · ⊂ Fd−1 /v ⊂ P/v
Let v0 be a vertex of ∆d and let φv : ∆d−1 = ∆d /v0 → P/v be a refinement homeomorphism that satisfies (i) and (ii) in P/v for the quotient
of the flag. This implies that there is a refinement homeomorphism from
star∆d (v0 ) = ∆d to pyrv (P/v), and a radial projection emanating from v
yields a refinement homeomorphism from ∆d to starP (v) = P .
To prove (i) and (ii) for this refinement, we need the following property
of radial projection.
27
F1
F2
F0
Figure 2.2: A refinement homeomorphism as constructed in the proof shown
for the star in the boundary of a 3-polytope. The differently shaded areas show
the images of the facets of ∆3 incident to the vertex that maps to F0 .
(A) Let F be a face of P that contains v. Then the image of a face
pyrv (F/v) of pyrv (P/v) under radial projection from v is the face F .
If Fi /v is a principal face of φv , then (A) implies that Fi is a principal
face of φ, for every i ∈ {0, . . . , d − 1}. This shows (i), as the statement for
i = −1 is trivial.
We now prove (ii). Fix i ∈ {−1, . . . , d} and let δ be an ℓ-face of ∆d
that contains φ−1 (Fi ). The statement is trivially true if i = −1, in which
case Fi = ∅. Otherwise, (ii) holds by induction in P/F0 = P/v for the
refinement φv . That is, the image φv (δ/v0 ) in P/v contains only (ℓ − 1)faces that contain Fi /v. The radial projection of such a face then contains
Fi by (A).
By induction, we can assume that the principal vertices of φv in P/v
are affinely independent. Consequently, the principal vertices of φ in P are
affinely independent.
See Figure 2.2 for an illustration of the main features of the refinement
homeomorphism constructed in the proof.
It is not true that the principal vertices are affinely independent in every refinement homeomorphism from the d-simplex to a d-polytope, and
examples illustrating this exist already in dimension 3: The regular 3crosspolytope for instance can be expressed as a refinement of the 3-simplex
28
2.1. Simplex Refinements of Polytopes
in which the equatorial vertices, that is, the four vertices with last coordinate zero, are principal.
Definition 2.1.3 (Rooted refinement). Let P be a d-polytope and v
a vertex of P . We say that a refinement homeomorphism with principal
vertices U ∪ {v} is rooted at v if U is a subset of the neighbors of v in G(P ).
Corollary 2.1.4. Let P be a d-polytope and v a vertex of P . Then there is
a refinement homeomorphism φ : ∆d → P that is rooted at v.
Proof. This follows from Theorem 2.1.2 (ii) for ℓ = 1.
Corollary 2.1.5 (Grünbaum [50]). Let P be a d-polytope, v ∈ V (P ) a
vertex of P , and G = G(P ) the graph of P . Then G contains a subdivision
of Kd+1 rooted at v.
Proof. The image of the graph of ∆d , which is a Kd+1 , under a refinement
homeomorphism as in Theorem 2.1.2 is a subdivision of Kd+1 rooted at the
vertex v.
As another exemplary application of Grünbaum’s theorem we derive the
existence of complementary flags in polytopal face lattices.
Corollary 2.1.6. Every flag has a complementary flag: For every flag
∅ = F−1 ⊂ F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fd−1 ⊂ Fd = P
there is a flag
∅ = G−1 ⊂ G0 ⊂ G1 ⊂ G2 ⊂ · · · ⊂ Gd−1 ⊂ Gd = P
such that Fi ∩ Gd−i−1 = ∅ and Fi ∪ Gd−i−1 is not contained in a facet, that
is, Fi and Gd−i−1 are complements in the face lattice of P , for every i with
−1 ≤ i ≤ d.
Proof. Let φ : ∆d → P be a refinement homeomorphism such that Fi is a
principal face for every i ∈ {−1, . . . , d}. Let
τ−1 = ∅ ⊂ τ0 ⊂ τ1 ⊂ τ2 ⊂ · · · ⊂ τd−1 ⊂ τd = ∆d
be the corresponding flag in ∆d , that is, τi = φ−1 (Fi ) for every i ∈ {−1, . . . , d}.
Take the (unique) complementary flag in ∆d . The image of this flag under
φ contains a flag in P that is complementary to F .
29
2.2
Lockeberg’s Counterexamples
Let φ : ∆d → P be a refinement homeomorphism, where P is a d-polytope.
We can imagine that this refinement amalgates facets of P into strongly
connected polytopal (d−1)-complexes—the images of the facets of ∆d under
the refinement. (Lockeberg calls such a complex a pseudofacet [69].)
We will interpret this amalgation as a coloring of the facets: Let m :=
fd−1 be the number of facets, let F1 , . . . , Fm be the facets of P , and let
τ1 , . . . , τd+1 be the facets of ∆d . Define a (surjective) map c : [m] → [d + 1]
by setting c(i) := j if φ(τi ) contains Fj . Thus, two facets Fj and Fk get the
same color i, that is, c(j) = c(k) = i, if and only if φ(τi ) contains both Fj
and Fk .
The intersection of the polytopal complexes corresponding to two (or
more) color classes is a polytopal complex in P . It is the image of the
intersection of the corresponding facets in ∆d under the refinement map.
Thus we can also deduce from this coloring the principal vertices (or
more general the principal faces, but we do not need this): Denote by
I(v) ⊆ [m] the indices of the facets that contain the given vertex v. A
vertex v is principal for the refinement φ if and only if c(I(v)) contains
exactly d colors.
In particular, if v is a vertex with |c(I(v))| = d, then there is no other
vertex w with c(I(v)) = c(I(w)). A violation of this condition is the main
contradiction we derive in the proof of the following theorem by Lockeberg [69].
Theorem 2.2.1 (Lockeberg [69, Theorem 6.1]). Let d ≥ 6. Then there
is a d-polytope P on d + 3 facets that has (necessarily nonadjacent) vertices
v1 and v2 that cannot both be principal in a refinement φ : ∆d → P .
Proof. Let P ∆ be the 6-polytope on 9 vertices given by the Gale diagram in
Figure 2.3 with vertex set [9] = {1, 2, 3, 4, 5, 6, 7, 8, 9}. By Theorem 1.3.3,
the complements of the sets {4, 5, 6} and {7, 8, 9} correspond to facets G1
and G2 of P ∆ , respectively.
Let P be the polar of P ∆ . We claim that the vertices v1 := G3
1 and
3
v2 := G2 , that is, the vertices of P that correspond under polarity to the
facets G1 and G2 , cannot both be principal in a simplex refinement of P .
Suppose φ : ∆d → P is a refinement homeomorphism with both v1 and
v2 principal. Let F1 , F2 , . . . , F9 be the facets of P , such that Fi corresponds
to the vertex i under polarity, for every i ∈ {1, . . . , 9}. Let c : [9] → [7] be
a coloring of the facets of P as described above.
30
2.2. Lockeberg’s Counterexamples
4 7
1
3
5 8
6 9
2
Figure 2.3: A Gale diagram of P ∆ . The polar polytope P is a polytope that
has two vertices that cannot be prescribed as principal vertices in a simplex
refinement of P . In contrast to every other Gale diagram in this thesis, here the
little number next to a vector is a label and not the multiplicity of that vector.
Since G1 and G2 are both simplex facets of P ∆ , the vertices v1 and v2 are
simple vertices of P . Then no two facets incident to v1 , or to v2 , can have
the same color. Thus, without loss of generality, we can assume c(1) = 1,
c(2) = 2, and c(3) = 3. Furthermore, there is a pair {i, j} with i ∈ {4, 5, 6}
and j ∈ {7, 8, 9} such that c(i) 6= c(j) and neither c(i) nor c(j) appears as
a color of any other facet.
Up to symmetry, we can assume that i = 4.
For j we cannot have j = 7: If j = 7, then the facets F1 , F4 and F7 must
intersect in a 3-face of P , as in this case all three of them would be principal
faces of the refinement. Consequently, the vertices 1, 4, 7 would have to lie
in a common 2-face of P ∆ . From the Gale diagram of P ∆ we infer that
the smallest face that contains 1, 4, 7 as vertices has as its complement the
set {3, 5, 8}. But this is a ridge of P ∆ , which is of dimension 5. Thus, up
to symmetry we can assume that j = 8. Without loss of generality we set
c(4) := 4 and c(8) := 7.
We are left, up to symmetry, with two possibilities of colors for the
remaining facets F5 , F6 , F7 , and F9 .
Case (i). Suppose c(5) = c(7) = 5 and c(6) = c(9) = 6. Consider the
facet of P ∆ given by {4, 5, 9}. Let w be the vertex of P that corresponds
to this facet. Then
I(v1 ) = {1, 2, 3, 7, 8, 9}
I(w) = {1, 2, 3, 6, 7, 8},
31
and we have
c(I(v1 )) = {1, 2, 3, 5, 6, 7} = c(I(w)),
so both v1 and w are principal with the same set of colors, which is a
contradiction.
Case (ii). Suppose c(5) = c(9) = 5 and c(6) = c(7) = 6. Let u and w be
the vertices of P that correspond to the facets of P ∆ given by {1, 6} and
{1, 9}. Then
I(u) = {2, 3, 4, 5, 7, 8, 9}
I(w) = {2, 3, 4, 5, 6, 7, 8}
and we have
c(I(u)) = {2, 3, 4, 5, 6, 7} = c(I(w)).
Again, this is a contradiction.
For d ≥ 7, let Q := pyrd−6 (P ). It is clear that the d − 6 pyramidal
vertices must be principal in any refinement φ : ∆d → Q, since any one of
them has exactly one facet that does not contain it. Let F be the (d − 7)simplex on the pyramidal vertices. Then any refinement of Q in which
v1 and v2 are principal induces a refinement of P in which v1 and v2 are
principal via the isomorphism P ∼
= Q/F .
A corresponding statement for pairs of higher dimensional faces is much
simpler to prove. For example, the bipyramid over a triangle has two edges
that cannot both be principal in a simplex refinement. By taking pyramids,
we can get examples for pairs of k-faces with k ≥ 2; compare the much
stronger statement in [69, Theorem 6.4].
The polytope constructed for Theorem 2.2.1 has a simple direct description without Gale diagrams; see Section 9.3.2 where we meet it again.
Lockeberg gave more counterexamples to the existence of refinement
homeomorphisms with certain properties [69]:
• For d ≥ 3, he has shown that the prism over a (d − 1)-simplex is a
counterexample to a statement proposed by Grünbaum [51, Exercise
11.1.4].
• For p, q ≥ 2, the polytope ∆p × ∆q has a set of three vertices that
are not principal for any Kd+1 -subdivision in G(∆p × ∆q ); see [69,
Proposition 5.1] and [46].
• For every d ≥ 4, there exists a simple d-polytope P with d + 4 facets
and v1 , v2 ∈ V (P ), such that P is not a refinement of ∆d with v1
and v2 principal. A polytope with this property can be constructed
by taking the polar of a polytope that was obtained from Cd (d + 3)
32
2.3. Refinements of Centrally Symmetric Polytopes
by addition of a vertex in suitable position (such a polytope appears
as an example of a neighborly 4-polytope that is not cyclic in [51,
p. 124]).
As for positive results, Lockeberg has shown the following:
• If P is a d-polytope on d + 2 facets, then for any two vertices of P
there is a refinement homeomorphism φ : ∆d → P with these two
vertices principal [69, Theorem 6.5].
• If P is a d-polytope with d ∈ {4, 5} and P has d + 3 facets, then
for any two vertices a refinement of ∆d with these two vertices as
principal vertices exists [69, Chapter 7].
Thus, with respect to the number of facets, the results and counterexamples
are best possible.
A polytope with the Gale diagram in Figure 2.3 is nonsimplicial, as it
has facets on d + 1 = 7 vertices. This suggests the following question.
Problem 2.2.2. Let P be a simplicial d-polytope, and let v1 , v2 ∈ V (P ).
Is there a refinement homeomorphism φ : ∆d → P with v1 and v2 both
principal?
This question has an affirmative answer if we restrict it to the 1-skeleton,
that is, every simplicial d-polytope contains a subdivision of Kd+1 with two
prescribed principal vertices. This was shown by Larman & Mani [66]; see
also Gallivan, Lockeberg & McMullen [46].
2.3
Refinements of Centrally Symmetric Polytopes
We make a very short trip into the land of centrally symmetric polytopes
that fits the topic of this chapter. Centrally symmetric polytopes, that is,
polytopes that are fixed by a central symmetry, do not play a role in the
rest of this thesis.
The following is immediate from the definition of refinement homeomorphisms.
Proposition 2.3.1. Let K and C be polytopal complexes and let φ : K → C
be a refinement homeomorphism. Then the f -vector of C is componentwise
larger than that of K, that is, fi (K) ≤ fi (C), with equality in all components
if and only if C and K are combinatorially isomorphic.
Indeed, for polytopes there are even stricter conditions. Because the
face lattice of a d-polytope is coatomic, the existence of a refinement home33
omorphism between two polytopes on the same number of facets implies
that these two polytopes are combinatorially isomorphic.
Definition 2.3.2 (Hanner polytope). A Hanner polytope is any polytope
that can be obtained by taking products and dualizing, starting from a line
segment.
We show here that the family of Hanner polytopes is not the finite
family asked for in [51, Exercise 11.1.5] if d ≥ 4, that is, Conjecture 2.0.8 is
false for d ≥ 4. However, Grünbaum remarks [51, Exercise 11.1.5], and this
was shown, for example, by Barnette [12], that Conjecture 2.0.8 is true in
dimension d = 3: Every centrally symmetric 3-polytope is the refinement of
a 3-cube or a 3-crosspolytope. Furthermore, the refinement can be chosen
to be compatible with the central symmetry.
Theorem 2.3.3. There are centrally symmetric d-polytopes for d ≥ 4 that
are not refinements of d-dimensional Hanner polytopes, that is, Conjecture 2.0.8 is false for d ≥ 4.
Proof. For d ≥ 5 this follows from recent results by Sanyal, Werner, and
Ziegler [99]. Specifically, they give examples of centrally symmetric polytopes for all d ≥ 5 for which no Hanner polytope with componentwise
smaller f -vector exists. These are counterexamples to a conjecture by
Kalai [61, Conjecture B].
In dimension d = 4, another example from [99] provides the example we
seek. Let
P := [−1, +1]4 ∩ {x ∈ R4 : −2 ≤ x1 + x2 + x3 + x4 ≤ 2}.
This polytope has f -vector f (P ) = (14, 36, 32, 10) [99] (the entries f0 = 14
and f3 = 10 are obvious). In dimension 4, there are four Hanner polytopes,
as summarized in the following table, see [99], where I := [0, 1] denotes the
unit interval.
polytope
(C3∆ × I)∆
C3∆ × I
C4
C4∆
f -vector
(10, 28, 30, 12)
(12, 30, 28, 10)
(16, 32, 24, 8)
(8, 24, 32, 16)
If P is a refinement of any of them, say H, then the f -vector of P
is componentwise larger than that of H. Thus, the only candidate for
H is C3∆ × I. However, this polytope and the polytope P both have 10
34
2.3. Refinements of Centrally Symmetric Polytopes
facets, but are not combinatorially isomorphic. Therefore, a refinement
homeomorphism cannot exist.
The following conjecture by Grünbaum [51, p. 216] would follow if
every simplicial centrally symmetric d-polytope is a refinement of the dcrosspolytope, according to a result by Halin [53].
Conjecture 2.3.4 (Grünbaum [51, p. 216]). The graph of every centrally symmetric simplicial d-polytope contains a refinement of the complete
graph on ⌊3d/2⌋ vertices.
Larman & Mani [66] have proven that every centrally symmetric simplicial 4-polytope contains a refinement of the complete graph on 6 vertices.
35
Chapter 3
Incidence Graphs of Polytopes
Balinski’s theorem [5] is a fundamental theorem on graphs of polytopes. It
states that the graph of every d-polytope is d-connected. It was reproved by
several other authors; see Barnette [6] [11], and Brøndsted & Maxwell [32].
There are various ways to generalize Balinski’s theorem, as was done
by Athanasiadis [3], Barnette [8] [10], Holt & Klee [54], Klee [64], Larman
& Rogers [67], Perles & Prabhu [90], and Sallee [98]. Björner [22] studied
higher topological connectivity of polytopes.
In this and the following chapter we are interested in the generalization
proposed by Sallee [98] and Athanasiadis [3]. Both consider the graph of a
polytope as a special case of the incidence graph on the faces of a polytope.
The (k, ℓ)-incidence graph Gk,ℓ (P ) is the graph that has as vertices the
k-faces of a given polytope P , and two vertices in this graph are connected
by an edge if and only if the corresponding k-faces lie in a common ℓ-face.
As a special case, we obtain the graph of a d-polytope for k = 0 and
ℓ = 1. Balinski’s theorem implies that this graph is d-connected. It is
also possible to derive from Balinski’s theorem that the (d − 2, d − 1)incidence graph is d-connected. This is the line graph of the dual graph of
the polytope.
Let d and k be nonnegative integers that satisfy 0 ≤ k ≤ d − 1, let P be
a d-polytope, and let Gk (P ) = Gk,k+1 (P ) be the (k, k + 1)-incidence graph
of P . Athanasiadis [3] has shown that
κ(Gk (P )) ≥
½
d,
(k + 1)(d − k),
37
if k = d − 2,
otherwise.
3. Incidence Graphs of Polytopes
This generalizes Balinski’s theorem to higher incidence graphs of polytopes.
Athanasiadis result is best possible, as there are d-polytopes that attain the
bound for all k and d.
Adding to this result, we determine in this chapter the connectivity of
(k, ℓ)-incidence graphs with ℓ ≥ 2k + 1. We show that in this case the
connectivity of the incidence graph Gk,ℓ (P ) of a d-polytope satisfies
¶
µ
d
.
κ(Gk,ℓ (P )) ≥
k+1
Again, this bound is best possible, as there are d-polytopes that attain the
bound for all d, ℓ and k with d − 1 ≥ ℓ ≥ 2k + 1.
We also briefly discuss a question by Athanasiadis, which asks which
(k, ℓ)-incidence graphs are graphs of polytopes.
Question 3.0.1 (Athanasiadis [4]). Which (k, ℓ)-incidence graphs are
polytopal, that is, (0, 1)-incidence graphs of other polytopes, not necessarily
of the same dimension?
We give a twofold answer for special choices of k and ℓ. We show that
(a) in every dimension d ≥ 3 there are infinitely many (1, 2)-incidence
graphs which are polytopal, and that (b) there are infinitely many nonpolytopal (0, 2)- and (1, 2)-incidence graphs of 3-polytopes. The main tool
for the construction of the examples is the truncating operation by Paffenholz & Ziegler [87].
3.1
Balinski’s Theorem for Polytopes
In this section we reprove Balinski’s theorem. Our version of this theorem
combines statements found in papers by Perles & Prabhu [90], and Holt &
Klee [54].
Definition 3.1.1. Let P be a d-polytope in Rd and let c be a linear function.
We denote by minc (P ) and maxc (P ) the sets of points of P for which c
attains its minimum and maximum, respectively.
The function c is generic with respect to P if for no two vertices u and
v of P the value c(u) equals the value c(v).
By Gc (P ) we denote the graph of P directed according to the values of
c, that is, (u, v) is a directed edge of Gc (P ) if and only if c(u) < c(v) and
(u, v) ∈ G(P ).
Theorem 3.1.2 (see Bondy & Murty [30, Theorem 11.6]). Let G
be a directed graph, and x and y two vertices of G. If for every set C ⊆
38
3.1. Balinski’s Theorem for Polytopes
V (G) \ {x, y} with |C| ≤ d − 1 there is a directed path from x to y in G \ C,
then there are d vertex-disjoint directed paths from x to y.
The following theorem combines statements by Perles & Prabhu [90],
and Holt & Klee [54]. The proof closely follows Holt & Klee’s proof [54].
Theorem 3.1.3 (“Monotone Balinski”). Let P be a d-polytope in Rd
and x, y ∈ V(P ). Let c be a linear function that is generic with respect to
P such that minc (P ) = x and maxc (P ) = y.
Let U ⊆ V(P ) \ {x, y} be a subset of the vertices with dim aff(U ) = e.
Then the following holds:
(i) If e = d − 1 and U is contained in some facet F of P , then there is a
monotone path in Gc (P ) \ U from x to y.
(ii) If e ≤ d − 2, then there are at least d − e − 1 monotone paths in Gc (P )
from x to y that only intersect in their endpoints x and y.
Proof. If d = 2 the statement is obvious. For d ≥ 3 we proceed by induction.
We first make a couple of assumptions on the position of P in Rd . We
assume that x = 0 is the origin. We also assume that the linear function
c is given by c(p) = pd , for p = (p1 , . . . , pd ) ∈ Rd , and that c(y) = 1. We
consider c as a row vector it has the form c = (0, 0, . . . , 0, 1).
We define a linear transformation on Rd by
p 7→ p + c(p)(ed − y).
Then y 7→ y + c(y)(ed − y) = ed . This transformation leaves the values of c
invariant, since
c(p) = c(p) + c(p) (c(ed ) − c(y)) = c(p + c(p)(ed − y)).
|
{z
}
=0
Thus, we can assume that y = ed = (0, . . . , 0, 1).
Case (i). Assume we have dim aff(U ) = e = d−1 and that U is contained
in some facet F of P . Let a be a vector that is normal to F and denote
by π the orthogonal projection of Rd onto the plane spanned by cT and a.
Then π(x) and π(y) are two vertices of the 2-dimensional polytope π(P ).
There is a path (v0 , e1 , v1 , . . . , en , vn ) in G(π(P )) that avoids the edge π(F )
and that is monotone with respect to cπ = (0, 1) = e2 . It remains to show
that this path lifts to a directed path in P that avoids F . This will be done
below.
Case (ii). Now, assume we have that e ≤ d − 2. Let U ′ be a set of
vertices in V \ (U ∪ {x, y}) with |U ′ | ≤ d − e − 2. Then for S = U ∪ U ′
39
we have that dim aff U ∪ U ′ ≤ e + (d − e − 3) + 1 = d − 2. According to
Theorem 3.1.2 it suffices to show that there is a directed path in Gc (P )
from x to y that avoids S.
Let J = {p ∈ Rd : pd−1 = 0}, J + be the positive half-space (pd−1 > 0)
bounded by J, and J − the negative half-space bounded by J. Denote by Φ
the orthogonal projection to the first d−1 coordinates. Then the projection
Φ(S) is contained in a (d − 2)-dimensional affine subspace H of J. Let H0
be the subspace parallel to H through the origin. By applying a further
transformation that leaves ed fixed, we can assume that H0 = {p ∈ Rd :
pd−1 = pd = 0} and that either S ⊆ J or S ⊆ J − .
Suppose that S ⊂ J − and that P does not intersect J + . Then J ∩ P is a
face F of P that contains x and y and a monotone path in Gc (F ). Clearly,
this path avoids S.
Now, either S ⊂ J − and P intersects J + or we have that S ⊂ J. In the
latter case we can assume that P intersects J + , otherwise we reflect in J.
Let π denote the projection to the plane spanned by ed−1 and ed . Then
π(x) and π(y) are vertices of a 2-dimensional polytope π(P ) that are the
maximum and minimum with respect to cπ = (0, 1) = e2 . There is a path
(v0 , e1 , v1 , . . . , vn−1 , en , vn ) in the boundary of π(P ) that avoids π(J − ) and
thus S.
It remains to show that the path v0 , e1 , v1 , . . . , en , vn constructed in the
two cases above can be lifted to a monotone path in P that avoids S.
The preimage π −1 (ei ) is a face Fi . Since in both cases the projection
preserves the values of c, that is, c(p) = cπ (π(p)), there are unique vertices
ui−1 and ui of Fi such that π(ui−1 ) = vi−1 and π(ui ) = vi , with v0 = x
and vn = y. Every Fi contains a monotone path connecting ui−1 and ui in
Gc (P ) that clearly avoids S. The union of these paths is the desired path
from x to y.
The following corollary can be derived by much simpler methods. For
example, the proof in [118, Theorem 3.14] also yields the following statement.
Corollary 3.1.4. Let P be a d-polytope and let C ⊆ V(P ) be a set of
vertices whose affine span is at most d − 2 dimensional. Then the graph
G(P ) \ C is connected.
Proof. Let x, y ∈ V(P ) be two vertices of P and let Hx and Hy be nonparallel supporting hyperplanes with Hx ∩ P = {x} and Hy ∩ P = {y}. Let H
be a third hyperplane with H ∩ P = ∅ and H ∩ Hx ∩ Hy = Hx ∩ Hy . Let
π be a projective transformation that sends H to infinity. Then π(Hx ) and
π(Hy ) are parallel hyperplanes, and, if c ∈ (Rd )∗ denotes a vector normal
40
3.2. Higher Incidence Graphs
to π(Hx ) and π(Hy ), we have that minc (π(P )) = x and minc (π(P )) = y.
By Theorem 3.1.3 with U = C, there is a monotone path from x to y.
Corollary 3.1.5 (Balinski’s theorem [5]). Let G be the graph of a dpolytope. Then G is d-connected.
Balinski’s theorem is best possible in the sense that for every f0 ≥ d + 1
there is a d-polytope on f0 vertices that is not (d + 1)-connected. For
instance, take a stacked polytope on f0 vertices, that is, a polytope obtained
from the simplex by iteratively stacking a d-simplex onto a facet f0 − d − 1
times. Such a polytope has a vertex of degree d, and so the graph either
has exactly d + 1 vertices, or there is a separating set of size d.
3.2
Higher Incidence Graphs
Definition 3.2.1. Let P be a d-polytope and let 0 ≤ k ≤ ℓ ≤ d. The
(k, ℓ)-incidence graph Gk,ℓ (P ) is the graph on the k-faces of P in which two
nodes are connected by an edge if they are contained in a common ℓ-face
of P . We write Gk (P ) for the (k, k + 1)-incidence graph of P .
A very simple example of a (k, ℓ)-incidence graph is the graph Gk,d ,
which is the complete graph on the k-faces of a polytope, that is, it is the
complete graph on fk vertices. The graph Gk,k is the complement of the
graph Gk,d .
An obvious but useful property is that Gk,ℓ is always a subgraph of Gk,ℓ′
for all ℓ ≤ ℓ′ ≤ d, with equality for ℓ + 1 ≤ ℓ′ ≤ d − 1 if P is ℓ′ -simplicial.
We define
κk,ℓ (d) := min{κ(Gk,ℓ (P )) : P is a d-polytope}.
Sallee [98] obtained various results on certain types of connectivities
related to (k, ℓ)-incidences in polytope skeleta, including bounds on the
connectivity of Gk,ℓ . He proved, see [98, p. 495], that
¶
µ ¶ µ
ℓ+1
ℓ
≤ κk,ℓ (d)
+
(d − 1 − ℓ)
k+1
k
(µ
¶ X
¶µ
¶)
ℓ−k µ
d
d−k
k+1
≤ min
,
.
k + 1 i=1
i
i
For k = 0, the bounds are equivalent to Balinski’s theorem. For larger k
the upper and the lower bound do not coincide.
41
G
F
v
Wv
Figure 3.1: The basic idea behind Athanasiadis’ proof.
¡ d ¢
The bipyramid over the (d − 1)-simplex gives the upper bound of k+1
Pℓ−k ¡d−k¢¡k+1¢
is
in the above inequalities, whereas the upper bound of i=1
i
i
given by the d-simplex. In the first case, the separating set consists of all
the k-faces of the base of the bipyramid. In the second case, the separating
set consists of all neighbors of a given k-face.
It is plausible that the upper bound is best possible, so one may conjecture that κk,ℓ (d) indeed equals the upper bound. This conjecture is supported by Athanasiadis’ recent result [3] that the graph Gk (P ) = Gk,k+1 (P )
is mk (d)-connected, where for k ≤ d − 1 we define
mk (d) :=
½
d,
(k + 1)(d − k),
k =d−2
otherwise.
Further support is given
¡ d ¢ by our result in the next section that the connectivity of Gk,ℓ (P ) is k+1 if ℓ ≥ 2k + 1.
As a preparation for the next section and for Chapter 4, where we prove
a conjectured generalization of Athanasiadis’ result [3, Conjecture 6.2], we
review the basic idea of Athanasiadis’ proof. One can find similar ideas in
Sallee’s work; see for example the first lemma in [98, Section 6].
Suppose that we are given two k-faces F and G of a polytope P . How
can we connect them by a path or a walk in Gk,k+1 (P )? Here is one way, as
is illustrated in Figure 3.1: Take a path W in the ordinary graph of P from
a vertex in F to a vertex in G. For every edge of this path, choose a k-face
that contains this edge. Two k-faces that contain a vertex v of the path can
be connected by a walk Wv in Gk,k+1 (P ) whose faces all contain the vertex
v. This follows by induction, as a walk in Gk−1,k (P/v) corresponds to such
a walk. The walks Wv now add up to a walk that connects F and G.
If we want to extend this idea to prove a higher connectivity of the
graph Gk,ℓ , we have to choose the path W with more care. Suppose that we
want to prove that Gk,ℓ (d) is f (k, ℓ, d)-connected. By definition, it suffices
42
3.3. Connectivity of Incidence Graphs
to show that, for every set of forbidden k-faces of size at most f (k, ℓ, d) − 1,
the graph Gk,ℓ without these forbidden faces is still connected.
This suggests the following two conditions on W :
¡d−1¢
forbidden k-faces. As there
• Every edge of W lies in less than k−1
¡d−1¢
are at least k−1 faces of dimension k that contain a fixed edge, this
condition implies that every edge has a free k-face, that is, a face that
does not lie in the set of forbidden faces.
• Every vertex of W does lie in at most f (k − 1, ℓ − 1, d − 1) of the
forbidden k-faces. Then it is possible to apply induction. Walks in the
graph Gk−1,ℓ−1 (P/v) of the vertex figure P/v at a vertex v correspond
to walks in Gk,ℓ (P ) that contain the vertex v.
This is the basic idea. It is put to work in the next section¡ with
¢
d
some variations, where we we prove that (k, ℓ)-incidence graphs are k+1
connected if ℓ ≥ 2k+1, and in the next chapter. There we prove a conjecture
by Athanasiadis on (k, k + 1)-incidence graphs of Cohen-Macaulay regular
cell complexes with intersection property.
It remains an open problem to determine the minimal connectivities for
(k, ℓ)-incidence graphs of polytopes for k + 2 ≤ ℓ ≤ 2k. It is likely that the
ideas and methods of this and the next chapter yield better bounds than
the ones proved by Sallee [98] also in this case.
Problem 3.2.2. What is the minimum connectivity κk,ℓ (d) of the (k, ℓ)incidence graphs for k + 2 ≤ ℓ ≤ 2k?
From what we know the answer could be
κk,ℓ (d) = min{κ(Gk,ℓ (∆d )), κ(Gk,ℓ (bipyr(∆d−1 )))}.
3.3
Connectivity of Incidence Graphs
In this section, we determine the connectivity of Gk,ℓ (P ) for ℓ ≥ 2k + 1. If
P = ∆d , then Gk,ℓ (P
) is
¢ the complete graph on the k-faces of ∆d , and we
¡d+1
have κ(Gk,ℓ )(P ) ≥ k+1 − 1.
Definition 3.3.1 (Good and bad vertices). Let P be a d-polytope
¡ and
¢
d
0 ≤ k ≤ d − 1. Let E be a set or multiset of k-faces of size less than k+1
.
¡d−1¢
We call a vertex v good if it is incident to at most k − 1 elements of E
(counting multiplicities if E is a multiset), and otherwise bad .
43
3.3.1
Neighborhoods in Incidence Graphs
Lemma 3.3.2. Let P be a d-polytope and ℓ ≥ 2k + 1. Then the minimum
degree δ of the (k, ℓ)-incidence graph satisfies
δ(Gk,ℓ (P )) ≥ δ(Gk,ℓ (∆d )).
Proof. Let F be a k-face of P and let φ : ∆d → P be a refinement as in
Theorem 2.1.2 such that F is a principal face (choose any flag that contains
the face F ). Let τ ⊆ ∆d with π(τ ) = F and let τ ′ be a neighbor of τ in
Gk,ℓ (∆d ) via the ℓ-face λ.
By Theorem 2.1.2, every ℓ-face in φ(λ) contains the k-face F . Thus,
every k-face in φ(τ ′ ) ⊆ φ(λ) is a neighbor of F in Gk,ℓ (P ). Since φ is
bijective, the statement follows.
Lemma 3.3.3. Let P be a d-polytope,
¡ d ¢ d ≥ k ≥ 0, and let E be a multiset
of k-faces of P of size less than k+1 . Then the bad vertices span at most
a (d − 2)-dimensional affine subspace of Rd .
Proof. Suppose their span would be at least (d − 1)-dimensional. Then
there is a set U of d affinely independent bad vertices. Any k-face in E
contains at most k + 1 of the vertices in U . But then double counting the
elements of E yields that
µ
¶
d−1
(k + 1)| E | ≥ d
,
k
and we have the contradiction
µ
¶ µ
¶
d−1
d
d
=
,
|E | ≥
k+1
k
k+1
which implies the statement.
Lemma 3.3.4. Let d ≥ ℓ ≥¡2k +
¢ 1 and E a multiset of k-faces of the dd
simplex ∆d of size less than k+1 . Let F be a k-face of ∆d that is not in
E. Then F contains a good vertex, or there is a k-face that is a neighbor of
F in Gk,ℓ (∆d ) that contains a good vertex.
Proof. Since ℓ ≥ 2k + 1, we have that the graph Gk,ℓ (∆d ) is the complete
graph on the k-faces of ∆d . The d+1 vertices of ∆d are affinely independent,
thus Lemma 3.3.3 implies that there are at most d−1 bad
¡d¢vertices in V(∆d ).
Let v ∈ V(∆d ) be one of the good vertices. There are k k-faces incident
to the good vertex v, so there is a k-face G with v ∈ G and G ∈
/ E.
44
Lemma 3.3.5. Let
¡ d d¢ ≥ ℓ ≥ 2k + 1 and E a set of k-faces of a d-polytope P
of size less than k+1 . Let F be a k-face that is not in E. Then F contains
a good vertex, or there is a k-face that is a neighbor of F in Gk,ℓ (P ) that
contains a good vertex.
Proof. Let φ : ∆d → P be a refinement homeomorphism as in Theorem 2.1.2 such that F is a principal face.
Define a multiset E ′ of the k-faces of the d-simplex as follows. One copy
of a k-face τ ⊆ ∆d is in E ′ for every k-face
¡ d ¢ of P in φ(τ ) ∩ || E ||. Since φ is
− 1.
bijective, we have that | E ′ | = | E | ≤ k+1
′
By Lemma 3.3.4, there is a k-face τ ⊆ ∆d that is not in E ′ and that
contains a good vertex v ′ . Then v := φ(v ′ ) is a good vertex in P .
Let λ be an ℓ-face that contains both τ and τ ′ . By Theorem 2.1.2 (ii),
the polytopal complex φ(λ) contains only ℓ-faces that contain F . Then
every k-face in φ(τ ) is a neighbor of F , and none of them lies in E, as
there is no copy of τ ′ in E ′ . Thus there is a k-face in φ(τ ′ ) with the desired
properties that contains the good vertex v.
3.3.2
Free Faces at Edges
Lemma 3.3.6. Let P be a d-polytope and suppose that d ≥ 2k + 1.
Assign a red label to some of the k-faces of P , and assign a green label
¡to dsome
¢ ¡ofd the
¢ k-faces of P such that each type of label is assigned at most
+ k−1 times (faces may have two labels of different colors).
k+1
Then there is a (2k + 1)-face that has one k-face that does not have a
red label and one k-face that does not have a green label.
Proof. We can assume that all faces either have a red label or a green label,
otherwise we pick an unlabeled k-face and any (2k + 1)-face that contains
that face.
There are
¶
¶ µ ¶ µ
¶ µ
¶ µ
µ
d
d
d
d
d+1
≥1
(3.1)
−
=
−
−
k−1
k
k−1
k+1
k+1
faces that only have a red label, as we have d ≥ 2k + 1. Let F be one of
them. All faces that lie in (2k + 1)-faces that contain F must also have
a red label, otherwise we have the desired result. By Lemma ¡3.3.2,
¢ the
d+1
. By
number of such k-faces is at least as many as in ∆d and this is k+1
the calculation given in Equation (3.1) at least one of them has only a green
label.
Recall the idea discussed in the last section that we want to apply to
prove the connectivity. Because of the huge number of forbidden k-faces,
45
we cannot necessarily always find a free k-face at every edge of a path in
the graph of a polytope, even if the two incident vertices are good.
The following lemma, however, shows that, if all the k-faces of an edge
with good endpoints are forbidden, then there is an edge in Gk,ℓ that connects a free k-face incident to one endpoint of the edge to a free k-face
incident to the other endpoint of the edge.
Lemma 3.3.7. Let P be a d-polytope,¡ d ≥
¢ ℓ ≥ 2k + 1 ≥ 3, and let E be a
d
. Furthermore, let e = vw be an
subset of the k-faces of size less than k+1
edge of P such that both endpoints v and w of e are good. Then
(i) there is a k-face G that is not in E that contains e, or
(ii) there is an ℓ-face G that has a free k-face incident to v and a free
k-face incident to w. Necessarily e is contained in G.
Proof. We assume that (i) does not hold, and prove (ii) under this assumption.
We want to apply Lemma 3.3.6 on the (k − 1)-faces of the edge figure
P/e at e. In the following we thus describe a labeling of these faces.
First, we label the (k − 1)-faces of P/v and of P/w. A (k − 1)-face of
P/v gets a green label, if the corresponding k-face in P lies in E. Otherwise,
it gets no label at all. Likewise, a (k − 1)-face of P/w gets a red label, if
the corresponding k-face in P lies in E. Otherwise, it gets no label at all.
Write v ′ for the vertex e/w in P/w and w′ for the vertex e/v in P/v.
We now label the (k − 1)-faces of P/e by “transporting” the labeling from
the vertex figures P/v and P/w to P/e. A (k − 1)-face F of P/e gets a
green label, if, under the natural isomorphism P/e ∼
= (P/v)/w′ , all the
(k − 1)-faces of the k-face of P/v that corresponds to F have a green label
according to the labeling in P/v.
As we also have a natural isomorphism between P/e and (P/w)/v ′ , the
same procedure can be applied to get a labeling with red labels: A (k − 1)face F of P/e gets a red label, if all the (k − 1)-faces of the k-face of P/w
that correponds to F have a red label according to the previous labeling.
We now count the number of faces in P/e that have a green label. By
symmetry, the same arguments apply for the number of faces with red
labels.
Consider a (k − 1)-face of P/e and suppose it has a green label. Let
F be the corresponding k-face of P/v. Then every (k − 1)-face of F has
a green label in P/v. Furthermore, since F corresponds to a face in P/e,
the face F contains the vertex w′ of P/v. Then there is a (k − 1)-face F ′
in ast(w′ , F ) with a green label that does not lie in any other k-face of P/v
that contains the vertex w′ .
46
This shows that the number of (k − 1)-faces with green labels of P/e
is at most the number of elements of E incident to v minus the number of
elements of E that contain the edge e.
Since
¡d−1¢v is a good vertex the number of faces of E incident to v is less
than k . Furthermore, since we assume that (i) does not hold, all k-faces
that contain e are in E.
Thus the number of k-faces in P/v all of whose (k − 1)-faces have green
labels is bounded by
¶
¶ µ
¶ µ
¶ µ
µ
d−2
d−2
d−1
d−1
.
+
=
−
k−2
k
k−1
k
The same bound holds for the number of faces with red labels.
We can therefore apply Lemma 3.3.6. This yields that there is a face of
dimension 2(k − 1) + 1 in P/e that has a (k − 1)-face that does not have
a green label and a (k − 1)-face that does not have a red label. This face
corresponds to an ℓ-face of P that has a k-face incident to v that is not in
E and a k-face incident to w that is not in E.
3.3.3
Proof of the Connectivity
Theorem 3.3.8. Let d ≥ ℓ ≥ 2k¡+ 1¢ and let P be a d-polytope. Then the
d
-connected.
(k, ℓ)-incidence graph Gk,ℓ (P ) is k+1
Proof. We prove the statement by induction on k. If k = 0 the statement
follows from Balinski’s theorem, Corollary 3.1.5, as G0,1 is a subgraph of
G0,ℓ for all ℓ ≥ 1 on the same set of vertices.
Thus,
¡ d ¢let k ≥ 1 and let E be a subset of the k-faces of cardinality less
. We have to show that Gk,ℓ (P )\E is connected, that is, for every
than k+1
pair F, G of k-faces not in E there is a path in Gk,ℓ (P ) \ E.
By Lemma 3.3.5, every k-face not in E has a neighbor that is not in
E that contains a good vertex. Thus, we can assume that F and G both
contain a good vertex. Let v be a good vertex of F and w a good vertex of
G.
By Lemma 3.3.3, the bad vertices affinely span at most a (d − 2)dimensional subspace of Rd . Thus, by Corollary 3.1.4 there is a walk W in
G(P ) that consists only of good vertices and that connects v to w.
Let u ∈ W be a vertex of this walk. Since ¡u is¢a good vertex, the
− 1. Furthermore,
number of k-faces in E that contain u is at most d−1
k
since ℓ ≥ 2k + 1 we have ℓ − 1 ≥ 2(k − 1) + 1. By the induction hypothesis,
any two (k − 1)-faces of P/u that correspond to faces that are not in E can
be connected in Gk−1,ℓ−1 (P/u) by a walk that avoids the (k − 1)-faces that
correspond to k-faces in E.
47
This path translates into a path of k-faces that contains u and avoids
the set E. Thus, any two k-faces that contain a vertex u of the walk W can
be connected by a path in Gk,ℓ (P ) \ E. The statement then follows from
Lemma 3.3.7.
The following lemma shows that the bound is best possible.
Lemma 3.3.9 (Sallee [98]). For every d ≥ 2 and 2k + 1 ≤ ℓ ≤ d¡ − 1¢
d
there is a d-polytope P such that Gk,ℓ (P ) can be separated by a set of k+1
k-faces.
Proof. Let P := bipyr(∆d−1 ) be
¢ bipyramid over the (d−1)-simplex with
¡ the
d
faces of dimension k in ∆d−1 separate
apexes x and y. Clearly, the k+1
the k-faces of P that contain x from the k-faces of P that contain y.
¡ d ¢
.
Corollary 3.3.10. Let 2k + 1 ≤ ℓ ≤ d − 1. Then κk,ℓ (d) = k+1
3.4
Polytopality of Incidence Graphs
We now consider a question on polytopality of incidence graphs. To avoid
confusion: We do not want to decide whether a given graph is a (k, ℓ)incidence graph of a d-polytope. Instead, we want to realize a given (k, ℓ)incidence graph as the (0, 1)-incidence graph of a different polytope.
Thus, this is our definition of polytopality.
Definition 3.4.1 (Polytopal). A graph G is d-polytopal if there is a dpolytope P with G(P ) = G. If a graph G is d-polytopal for some d, we say
that G is polytopal.
It was observed by Athanasiadis [4] that there are indeed (k, ℓ)-incidence
graphs that are also (0, 1)-incidence graphs, that is, that are polytope in
the sense of Definition 3.4.1
Take, for example, the d-simplex ∆d on d + 1 vertices. If ℓ ≥ 2k + 1,
then any two k-faces F1 and F2 have a common ℓ-face G that ¡contains
them
¢
d+1
vertices.
both. Thus, in this case Gk,ℓ (∆d ) is the complete graph on k+1
¡d+1¢
− 1. If ℓ = 2k = d − 1,
This graph is d′ -polytopal for every 4 ≤ d′ ≤ k+1
any two k-faces F and G of Gk,ℓ (∆d ) are connected by an edge, unless
| V(F1 ) ∪ V(F2 )| = 2k + 2 = d + 1. Thus, for every k-face F there is exactly
one k-face
¡d+1¢G that is not a neighbor of F . The graph Gk,ℓ is Km \ E, where
m := k+1 and E denotes a perfect matching in Km . This is the graph of
a crosspolytope of suitable dimension.
In this section, we discuss this question in more detail and look at some
interesting examples.
48
3.4. Polytopality of Incidence Graphs
(a) The graph of the 3-crosspolytope
is the (1, 2)-incidence graph of the 3simplex.
(b) Example of a polytope with a nonpolytopal (0, 2)-incidence-graph.
The
graph G0,2 clearly is not 4-connected, but
has a K6 -subgraph.
Figure 3.2: Polytopal and nonpolytopal incidence graphs.
3.4.1
Polytopal Incidence Graphs
To construct infinitely many polytopal (1, 2)-incidence graphs we look at
Paffenholz & Ziegler’s truncatable polytopes [87].
Definition 3.4.2 (Truncatable Polytopes). If P is a d-polytope, then
P is called truncatable if for every edge e there is a relative interior point
ve such that the polytope T (P ) := conv{ve : e ∈ E(G(P ))} is obtainable
by truncating all vertices of P , that is, by cutting off all vertices by suitable
hyperplanes.
The following proposition is “geometrically clear.”
Proposition 3.4.3. Let P be a 2-simplicial truncatable d-polytope. Then
we have that G0,1 (T (P )) ∼
= G1,2 (P ).
See Figure 3.2(a) for a realization of T (∆3 ), which is an octahedron, as
a truncation of ∆3 .
According to the following theorem by Paffenholz & Ziegler [87], truncatable polytopes can be constructed by finding realizations such that all
edges of the polytope are tangent to a sphere.
Theorem 3.4.4 (Paffenholz & Ziegler [87]). Let P be a d-polytope for
d ≥ 3 such that all edges of P are tangent to a (d − 1)-sphere. Then P is
truncatable.
49
The case of dimension 2 is not interesting: Every 2-polytope can be
realized with all edges tangent to a 1-sphere. This is trivial.
In dimension 3, the analogous statement holds but is far from trivial.
Theorem 3.4.5 (Koebe-Andreev-Thurston Theorem; see [118, Theorem 4.12]). Every 3-polytope can be realized with all edges tangent to some
2-sphere.
This theorem is proved by non-linear methods, for example, the variational principles by Bobenko & Springborn [27]; see also Ziegler [120].
For 3-polytopes we can consequently describe exactly when a (k, ℓ)incidence graph is polytopal. This is done in the following theorem.
Theorem 3.4.6. Let P be a 3-polytope and k ≤ ℓ + 1 with k ∈ {0, 1, 2} and
ℓ ∈ {1, 2, 3}. Then the following hold:
(i) If k = 0, then Gk,ℓ is 3-polytopal if and only if P is ℓ-simplicial.
(ii) If k = 1, then Gk,ℓ is 3-polytopal if and only if P is simplicial.
(iii) If k = 2, then Gk,ℓ = G2,3 = Kf2 is 3-polytopal if and only if P is the
3-simplex.
Proof. Let k = 0. If ℓ = 1, then G trivially is 3-polytopal, so assume
ℓ ∈ {2, 3}. If ℓ = 3, P cannot have more than 4 vertices, so P must be the
3-simplex. If ℓ = 2 and P is 2-simplicial, then G0,2 = G2,3 . Otherwise, P
has an n-gon face for some n ≥ 4. Then G0,2 clearly has more edges than
a triangulated planar graph and thus is not planar. This shows (i).
Suppose k = 1. If P is simplicial, then G1,2 (P ) ∼
= G0,1 (T (P )). If P is
not simplicial, then G1,2 is not planar for the same reason as G0,2 is not
planar. Thus (ii) follows.
Finally, if k = 2 we have Gk,ℓ = Kf2 . Since every 3-polytope that is not
the simplex has more than 4 faces of dimension 2. Thus (iii) is proved.
For higher dimensions, “edge tangency” may fail. However, this property is not necessary for truncatability. We have the following theorem by
Paffenholz & Ziegler [87].
Theorem 3.4.7 (Paffenholz & Ziegler [87, Theorem 3.5]). Let P be a
stacked d-polytope and d ≥ 3. Then P has a realization that is truncatable.
This yields an infinite family of polytopal (1, 2)-incidence graphs of simplicial polytopes in fixed dimension.
Theorem 3.4.8. For every d ≥ 3 there is an infinite family of simplicial
d-polytopes with polytopal (1, 2)-incidence graphs.
50
3.4. Polytopality of Incidence Graphs
3.4.2
Nonpolytopal Incidence Graphs
We exploit two conditions of polytope graphs to show that there are infinite series of nonpolytopal (0, 2)- or (1, 2)-incidence graphs of 3-polytopes,
namely that (a) a graph of a 3-polytope does not have a K5 subgraph, and
that (b) a graph of a d-polytope with d ≥ 4 is at least d-connected by
Balinski’s theorem.
Theorem 3.4.9. There is an infinite number of combinatorial types of 3polytopes such that neither G0,2 nor G1,2 is d-polytopal for any d ∈ N.
Proof. Consider the (0, 2)-incidence graph G of the polytope given in Figure 3.2(b). The subgraph that corresponds to the 5-gon face is a K5 . Thus,
G is not the graph of a 3-polytope. However, it is also not the graph of
a polytope of larger dimension, as it is clearly not 4-connected. The same
polytope also yields an example of a (1, 2)-incidence graph that is not the
graph of a polytope.
Starting from this example, we clearly can construct an infinite family
of nonplanar (0, 2)- or (1, 2)-incidence graphs that are at most 3-connected
by stacking onto triangular faces.
For higher dimensions, finding nonpolytopal incidence graphs might be
an intractable problem, as every graph is an induced subgraph of a 4polytope; see [63, Proposition 20.2.3].
51
Chapter 4
Athanasiadis’ Conjecture on
Incidence Graphs
Athanasiadis conjectured that his result on (k, k + 1)-incidence graphs that
we briefly discussed in the last chapter would extend to a certain class of
cell complexes, Cohen-Macaulay regular cell complexes with intersection
property [3, Conjecture 6.2].
We prove his conjecture by establishing the connectivity of (k, k + 1)incidence graphs of a subclass of Barnette’s graph manifolds [8]. It is then
easy to extend this result to weakly normal d-graph complexes, which we
define in Section 4.3. These are a combinatorial generalization of CohenMacaulay regular cell complexes with intersection property.
The reader acquainted with Athanasiadis’ proof might have a feeling of
déjà-vu in this chapter: The proof given here is basically Athanasiadis’ proof
adapted to the more general setting. In the end I have tried to preserve as
much of this parallelism as possible, although originally I had not planned
to do so.
I do not know if the result on (k, ℓ)-incidence graphs with ℓ ≥ 2k + 1
that we proved in Chapter 3 also generalizes to the setting of this chapter.
In the proof of this result we had used the fact that every d-polytope is a
refinement of the d-simplex, and I do not know how to generalize this to
graph manifolds in a meaningful way.
53
4. Athanasiadis’ Conjecture on Incidence Graphs
4.1
Regular Cell Complexes
The definitions and notation for regular cell complexes are from [23, Section 4.7].
Definition 4.1.1 (Regular cell complex). A regularScell complex C is a
finite collection of balls σ in a Hausdorff space k C k = σ∈C σ that satisfies
the following conditions:
(i) The empty set ∅ is a member of C,
◦
(ii) the interiors σ partition k C k, and
(iii) the boundary ∂σ of every σ is the union of some members of C, for all
σ ∈ C.
We call the balls σ the cells or faces of C. The dimension of a regular
cell complex is the maximal dimension of one of its faces.
Faces of dimension 0 and 1 are called the vertices and edges of C, respectively. Denote by f0 (C) the number of vertices of C. We write V(C) for
the set of vertices of C, and G(C) for the graph of C, that is, the regular cell
complex given by the vertices and edges of C.
If C is d-dimensional, then the faces of dimension d are called facets, and
the faces of dimension d − 1 are called ridges.
If T is a topological space and T = k C k then C is called a regular cell
decomposition of T . The set of all faces σ ∈ C ordered by inclusion forms a
poset, the face poset F(C).
A regular cell complex C is Cohen-Macaulay if its face poset F(C) is a
Cohen-Macaulay poset; see [21, Section 11]. We only need the following
two properties of Cohen-Macaulay cell complexes. They are pure, that is,
every face lies in a facet, and they are strongly connected, that is, every two
facets can be connected by a sequence of facets in which two consecutive
facets intersect in a ridge.
A regular cell complex is strongly regular if it satisfies the intersection
property, that is, the intersection of two faces is a single face.
4.2
Athanasiadis’ Conjecture
Given a strongly regular cell complex C, we denote by Gk (C) the graph on
the k-cells of C in which two k-cells are connected by an edge if they lie on
a common (k + 1)-cell. Thus, we have G0 (C) = G(C).
Athanasiadis conjectured the following extension of his result on the
connectivity of (k, k + 1)-incidence graph of polytopes.
54
4.2. Athanasiadis’ Conjecture
Conjecture 4.2.1 (Athanasiadis [3, Conjecture 6.2]). Let C be a ddimensional Cohen-Macaulay strongly regular cell complex. Then the graph
Gk (C) is
• (k + 1)(d − k)-connected if 0 ≤ k ≤ d − 3 and
• d-connected if k = d − 2.
(For k = d − 1, the generalization is false: Consider, for example, the
complex consisting of two d-simplices glued together along a common facet.)
The main difficulty in adapting the proof is that face figures of regular
cell complexes are not regular cell complexes, in general. Consider, for
instance, the double suspension of the Poincaré homology 3-sphere [35] [92].
To circumvent this problem we consider Conjecture 4.2.1 in greater generality in the setting of Barnette’s graph manifolds [8]. We have a similar
problem here; a face figure of a graph manifold is not necessarily a graph
manifold. The punchline will be however that the class of graph manifolds that is closed under taking vertex figures is still large enough to cover
Athanasiadis’ conjecture. There are some technical subtleties though, as
discussed in Section 4.3.
With the right setup and definitions we have that the strong components of a vertex figure of a graph manifold are graph manifolds. The only
technical problem then is to control the different components of the vertex
figures. This is done using the notion of strong walks, which is introduced
in Section 4.4.2.
A strong walk is, basically, a walk in the graph of a graph manifold
along a strong chain of facets, that is, along a sequence of facets such that
consecutive facets intersect in a ridge. Because a strong chain cannot jump
between strong components in a face figure, a strong walk “selects” strong
components in a consistent way.
There is an alternative approach that might turn out to be more elegant:
A normalization procedure described by Stanley [106, p. 83], Goresky &
MacPherson [47, p. 151], Björner [19], and Kalai [59], turns a simplicial
pseudo-manifold into a normal one, that is, one in which links of faces
are connected. This technique works well in connection with lower bound
theorems; see Kalai [59] and Tay [111]. I do not know how to make it work
for the connectivity problem, though. The problem is that disjoint paths in
an incidence graph of the normalized manifold are not necessarily disjoint
in the manifold we started with (but see [10]).
55
4.3
Graph Manifolds
Graph manifolds were devised by Barnette as a tool to study graph properties of manifolds [8].
Definition 4.3.1 (Face structure, graph manifold, pseudo-graph
manifold). A d-face structure M is a graph, together with a collection C
of r-face structures, for −2 ≤ r ≤ d − 1, called the faces of M (we also
call M a face), inductively defined as follows. A (−2)-face structure is the
empty set. A (−1)-face structure is single vertex v with faces v and ∅. A
0-face structure is an edge e = uv with faces e, u, v, and ∅.
For d ≥ 1, the collection C of faces of the d-face structure M is a
collection of r-face structures, for −2 ≤ r ≤ d−1 that satisfies the following
conditions:
(1) If F ∈ C is a face of M, and F ′ is a face of F , then F ′ ∈ C (closed
under taking faces).
(2) Every face F of M is a face of a (d − 1)-face structure of M (pure).
If F is an r-face structure and F ∈ C, we say that F is an r-face of M.
The 0-, 1-, (d − 2)-, and (d − 1)-faces of M are the vertices, edges, ridges,
and facets of M, respectively. We denote the vertices of M by V(M).
We say that M
(3) has the intersection property if for all F, G ∈ C the intersection F ∩ G
is a face of F and of G;
(4) is strongly connected if for every two facets F, F ′ ∈ C there is a sequence
F = F 0 , R1 , F 1 , R2 , . . . , Rn , F n = F ′
of facets and ridges, called a strong chain, such that F i ∩ F i+1 = Ri+1 ,
for all i = 0, . . . , n − 1;
(5) has the pseudo-manifold property if each ridge of M is a face of exactly
two facets of M.
A weak d-graph manifold is a d-face structure that satisfies (3) and (5),
and such that every r-face of M, for −2 ≤ r ≤ d − 1, is again a weak
r-graph manifold. For example, a weak 1-graph manifold is a collection of
disjoint cycles and all their faces.
A d-graph manifold is a weak d-graph manifold that additionally satisfies
(4), and such that every r-face, −2 ≤ r ≤ d−1, is again an r-graph manifold.
56
4.3. Graph Manifolds
face structure
(1, 2)
weak graph manifold
(3, 5)
pseudo-graph manifold
(faces are graph manifolds)
graph complex
(3, 4)
graph manifold
weakly normal
graph complex
weakly normal
graph manifold
normal graph complex
normal graph manifold
Figure 4.1: Inclusion relations among some of the structures defined in this
section.
A d-pseudo-graph manifold is a weak d-graph manifold such that every
r-face, −2 ≤ r ≤ d − 1, is an r-graph manifold.
If a d-face structure M satisfies (3) and (4), and every r-face of M,
for −2 ≤ r ≤ d − 1, is an r-graph manifold, we say that M is a d-graph
complex.
A d-graph complex such that
(6) each ridge of M is contained in at most two facets of M
is a d-graph manifold with boundary.
The definitions of graph manifold and pseudo-graph manifold coincide
with the definitions given by Barnette [8].
The defined structures are related by their inclusions in Figure 4.1.
There already the classes of weakly normal and normal graph manifolds
appear. We will define these classes, as well as the corresponding graph
complexes, below.
57
Definition 4.3.2 (Face figure, cf. [8, Proof of Theorem 7]). Let M
be a d-graph manifold, and F a (k − 1)-face of M, for 0 ≤ k ≤ d. The face
figure of M at F , denoted M /F , is the (d − k − 1)-face structure obtained
in the following way:
(i) For every k-face in M that contains F there is a vertex in M /F .
(ii) Inductively, a collection of r-faces of M /F , for −1 ≤ r ≤ d − 3,
determines an (r + 1)-face of M /v if and only if the corresponding
(r + k + 1)-faces of M are exactly the (r + k + 1)-faces of an (r + k + 2)face of M that contain F .
If F is a vertex, we call M /F a vertex figure, and if F is an edge, we call
M /F an edge figure.
The poset of faces of a graph manifold is a graded lattice, with M as 1̂
and ∅ as 0̂. Every interval in this lattice is the face poset of a weak graph
manifold and face figures are upper intervals in this lattice.
A subtle mistake seems to appear in Barnette’s work [8, Proof of Theorem 7] in connection with vertex figures of graph manifolds. He claims
that a vertex figure “clearly” is a pseudo-graph manifold [8, p. 67] and
concludes that the maximal strongly connected components of the vertex
figure, called the strong components, are graph manifolds. Indeed, it follows
from properties (1), (2), (3), and (5) that these also hold for face figures.
Thus, by induction a face figure of a graph manifold (or more generally,
of a weak graph manifold) is a weak graph manifold. However, a face of a
face figure may fail to be a graph manifold. Thus, Barnette’s claim that a
vertex figure is a pseudo-graph manifold is incorrect.
To support our point, we construct an explicit example of a graph manifold in which the unique strong component of a vertex figure is not a graph
manifold.
Given a graph manifold M we construct a graph manifold pyr(M),
called the pyramid over M, in the following way: The graph of pyr(M) is
the join of the graph of M with an additional vertex v0 , that is, it consists
of the graph of M and all the edges between v0 and vertices of the graph of
M. The pyramid over a (−2)-graph manifold, that is, the empty set, is the
single vertex v0 with v0 and ∅ as its faces. For higher dimensions, all the
faces of M are faces of pyr(M), and for every face F of M, the pyramid
pyr(F ) is a face of pyr(M). If M is a d-graph manifold, then pyr(M) is a
(d + 1)-graph manifold.
The 2-graph manifold M in Figure 4.2(a) (after a figure by Barnette [10])
is our starting point for our counterexample. The vertex figure at v is a
58
4.3. Graph Manifolds
v
F
v0
(a) A pinched torus. The vertex figure at
v consists of two disjoint cycles.
(b) The vertex figure at v of the pyramid
over the pinched torus. The face F consists of two disjoint cycles.
disjoint union of two triangles, which is a weak graph manifold, and the
strong components of it are graph manifolds. So, it is indeed a pseudograph manifold. However, if we take M′ as the pyramid over M, then the
vertex figure at v of M′ is the weak graph manifold in Figure 4.2(b) (where
v0 is the vertex of M′ /v that corresponds to the edge from v to the pyramid
apex).
What is strange about this weak 2-graph manifold? The two disjoint
cycles that form the base of the two pyramids with apex v0 form a single
face F , according to Definition 4.3.2. In particular, this vertex figure has
only one strong component, as a strong chain can “jump” via F from one
pyramid to the other.
Barnette’s proof of [8, Theorem 7] by induction via vertex figures is
rendered invalid by this problem. The result of this theorem that every
d-graph manifold contains a subdivision of Kd+1 can however be proved
for weak d-graph manifolds with exactly the same proof. This implies the
result for d-graph manifolds. So, in this case, it is just a matter of choosing
the right generality for the induction.
For our methods in this chapter it is however crucial that the strong
components of a face figure are graph manifolds. Thus we define the class
of weakly normal graph manifolds as follows.
Definition 4.3.3. A d-graph manifold is weakly normal if either d ≤ 1 or
d ≥ 2 and every strong component of every vertex figure is again a weakly
normal graph manifold. It is normal if either d ≤ 1 or d ≥ 2 and every
vertex figure is a normal graph manifold.
59
By this definition, all 2-graph manifolds are weakly normal, but not
all are normal. For example, the pinched torus in Figure 4.2(a) is weakly
normal, but not normal. Another example is given by the collection of
graphs of the faces of a d-polytope. They form a normal (d − 1)-graph
manifold.
Definition 4.3.4. A d-graph complex is weakly normal if every face of it is
a weakly normal graph manifold. It is normal if every face of it is a normal
graph manifold.
The generality of normal graph manifolds is still sufficient for a proof
of Athanasiadis’ conjecture. Nevertheless, we prove the more general result
for weakly normal graph complexes.
Definition 4.3.5 (Antistar, star, link). Let M be a weak graph manifold
and F a face of M.
Define the d-face structure ast(F, M), the antistar of M at F , to be
the collection of all faces of M that do not contain F .
Define the d-face structure star(F, M ), the star of M at F , to be the
collection of all faces that are contained in a face that contains F .
Finally, define the link of M at F by link(F, M ) := ast(F, M )∩star(F, M ).
This is a (d − 1)-face structure.
4.4
Basic Properties of Graph Manifolds
¡d+2¢
faces
In this section, we (a) prove that a d-graph manifold has at least k+2
of dimension k, (b) introduce the notion of strong walk, and (c) prove two
variants of Balinski’s theorem for graph manifolds.
4.4.1
Face Numbers of Graph Manifolds
Lemma 4.4.1. Let M be a weak d-graph manifold with d ≥ 0 and let u, v
be vertices of M. Then there is a facet that contains u but not v. In
particular, the antistar of any vertex contains at least one facet.
Proof. If M is a weak 0-graph manifold the statement clearly holds. Let
M be a weak d-graph manifold with d ≥ 1, and let u, v ∈ V(M). Let F
be a facet that contains u. If F does not meet v, we are done. Otherwise,
by induction, there is a ridge R in F (a facet of the facet F) that contains
u but not v. By the pseudo-manifold property (5), there is a facet F ′ ,
distinct from F, that contains R. By the intersection property (3) we have
F ∩ F ′ = R. Thus, F ′ is a facet that contains u but not v.
60
4.4. Basic Properties of Graph Manifolds
Lemma 4.4.2.
¡d+2¢Let d ≥ −2 and M be a weak d-graph manifold. Then M
has at least k+2 faces of dimension k, for all −2 ≤ k ≤ d.
Proof. We prove the statement by induction on the dimension. The cases
d = −2, −1 and k = −2, d are trivial.
Let M be a weak d-graph manifold, where d ≥ 0, and let −1 ≤ k ≤ d−1.
Let v be a vertex of M and choose a facet F in ast(v, M), which
¡d+1¢exists
according to Lemma 4.4.1. By induction, the facet F has at least k+2 faces
of dimension k. These are k-faces of M that do not contain v. ¡Additionally,
¢
d+1
faces of
any strong component of the vertex figure at v has at least k+1
dimension k − 1. These correspond to k-faces of M that contain v. Thus,
in total M has at least
¶
¶ µ
¶ µ
µ
d+2
d+1
d+1
=
+
k+2
k+1
k+2
faces of dimension k.
Corollary 4.4.3. A d-graph manifold has at least
k, for all −2 ≤ k ≤ d.
4.4.2
¡d+2¢
k+2
faces of dimension
Strong Walks in Graph Manifolds
If C 1 and C 2 are strong chains in a graph manifold, and the last facet of C 1
coincides with the first facet of C 2 we write C 1 C 2 for the concatenation of
C 1 and C 2 , that is, for the strong chain that first traverses C 1 and then C 2 .
Definition 4.4.4 (Strong walk). Let M be a weakly normal graph manifold. A strong walk in M is a pair (W, χ) consisting of a walk
W = (v0 , e1 , v1 , . . . , en , vn )
in G(M) and a map χ that associates to every i ∈ {0, . . . , n} a strong chain
χ(i) such that
• vi ∈ F for every facet F in χ(i), and
• the last facet of χ(i) equals the first facet of χ(i + 1) for every i ∈
{0, . . . , n − 1}, that is, the concatenation C χ := χ(0)χ(1) · · · χ(n) is a
strong chain in M.
A strong walk (W, χ) with W = (v0 , e1 , v1 , . . . , en , vn ) determines for every i ∈ {0, . . . , n} a strong component of M /vi and for every i ∈ {1, . . . , n}
a strong component of M /ei .
61
F0
Fn
Vi
vi
W
C χ = (F 0 , . . . , F m )
Figure 4.2: A strong walk (W, χ) with the strong component V i induced at i.
We denote the strong component of M /vi that contains the quotient
of the first facet of χ(i) with vi by V i for every i ∈ {0, . . . , n}. Then M /vi
contains every quotient of a facet of χ(i) with vi , as χ(i) is a strong chain
of facets that all contain vi .
Furthermore, for every i ∈ {1, . . . , n} we have that the first facet of χ(i)
contains the edge ei , because the first facet of χ(i) must coincide with the
last facet of χ(i − 1) and consequently must contain both vi−1 and vi . We
denote the strong component of M /ei that contains the quotient of the
first facet of χ(i) with ei by E i for every i ∈ {0, . . . , n}.
It is important that the vertices ei /vi ∈ V(M /vi ) and ei+1 /vi ∈ V(M /vi )
both lie in V i and that a k-face of E i corresponds to a (k + 1)-face in V i−1
and to a (k + 1)-face in V i .
Definition 4.4.5 (Concatenation of strong walks). Let (W, χ) and
(W ′ , χ′ ) be two strong walks in M such that
• the last vertex of W and the first vertex of W ′ coincide,
• the last facet of C χ and the first facet of C χ′ coincide.
Suppose that the lengths of W and W ′ are n and n′ , respectively.
The concatenation of (W, χ) and (W ′ , χ′ ) is the strong walk
(W, χ)(W ′ , χ′ ) := (W W ′ , χ̃),
where W W ′ is the usual concatenation
defined by

for
 χ(i),
χ(i)χ′ (0),
for
χ̃(i) :=
 ′
χ (i − n),
for
62
of walks in graphs, and where χ̃ is
i ∈ {0, . . . , n}
i=n
i ∈ {n + 1, . . . , n + n′ }.
4.4. Basic Properties of Graph Manifolds
We describe how to construct a strong walk from a vertex v to a vertex
w if we are given a strong chain C = (F 0 , R1 , F 1 , . . . , Rm , F m ) with v ∈ F 0
and w ∈ F m . Let w0 := v, wi a vertex of Ri for i ∈ {1, . . . , m − 1}, and
wm = w. Choose a walk Wi from wi to wi+1 in F i for every i ∈ {0, . . . , m},
and let W = W0 W1 · · · Wm .
Write Wi as (v0 , e1 , v1 , . . . , en , vn ), that is, we have v0 = wi and vn =
wi+1 . Define a strong chain (Wi , χi ) by setting χi (j) = F i for every j ∈
{0, . . . , n − 1}. If i 6= m, we set χi (n) = (F i , Ri , F i+1 ), otherwise χi (n) =
F i = F m.
Then
(W, χ) := (W0 , χ0 )(W1 , χ1 ) · · · (Wm , χm )
is a strong walk such that the walk W connects the vertices v and w.
4.4.3
Balinski’s Theorem for Graph Manifolds
Lemma 4.4.6 (Barnette [8, Lemma 2]). The antistar of any vertex of
a d-graph manifold is a d-graph manifold with boundary. In particular, it
is strongly connected.
Theorem 4.4.7 (Barnette [8]). Let M be a d-graph manifold. Then the
graph of M is (d + 1)-connected.
Furthermore, if S ⊆ V(M) is a set of vertices of cardinality | S | = d
that separates a strong chain, then S is contained in a facet of M.
Proof. The graph manifold M has at least d + 2 vertices by Corollary 4.4.3.
Let S be a subset of the vertices of M of cardinality at most d. Let
u ∈ S and choose v, w ∈ V(M )\S. Let F and F ′ be facets of ast(u, M) with
v ∈ F and w ∈ F ′ . These exist by Lemma 4.4.1. By Lemma 4.4.6, there
is a strong chain of facets F = F 0 , R1 , F 1 , . . . , Rn , F n = F ′ in ast(u, M).
By induction, the graph of every F i is d-connected. Furthermore, every
intersection Ri+1 , i ∈ {0, . . . , n − 1}, is a (d − 2)-graph manifold and thus
has at least d vertices, by Corollary 4.4.3.
Thus, if we choose vertices w1 , . . . , wn with wi ∈ V(Ri ) \ S and set
w0 := v and wn+1 := w, there are walks W0 ⊆ F 0 , . . . , Wn ⊆ F n such that
Wi connects wi to wi+1 .
Thus, there is a walk W = W0 W1 · · · Wn starting in v and ending in w
that avoids S. Clearly, if a strong chain F ′0 , R′1 , F ′1 , . . . , R′n , F ′n is separated
by S, then either S separates the graph of one facet F ′i and we have S ⊆ F ′i ,
or all vertices in S lie in some intersection R′i+1 = F ′i ∩ F ′i+1 .
Lemma 4.4.8. Let d ≥ −1 and M be a d-graph manifold. Then G(M) \
V(F) is connected for any facet F of M.
63
Proof. We prove the statement by induction. The case d = −1 is trivial.
Let F be a facet of M and u ∈ V(F) a vertex of F. Let v, w be two
vertices in G(M) \ V(F). By Lemma 4.4.1, there are facets G 1 and G 2
with v ∈ G 1 and w ∈ G 2 that avoid u (possibly G 1 = G 2 ). Clearly, G 1
and G 2 lie in ast(u, M). By Lemma 4.4.6 there is a strong chain of facets
G 1 = F 0 , R1 , F 1 , . . . , Rn , F n = G 2 that connects G 1 and G 2 in ast(u, M).
By induction, G(F i ) \ V(F i ∩ F) is connected for every i ∈ {0, . . . , n},
as we can choose a facet that contains F i ∩ F. Furthermore, for every
i ∈ {1, . . . , n} we have Ri \ F =
6 ∅, as Ri is not contained in F. Thus, there
is a walk in G(M) \ V(F) that connects v, w.
Similarly one shows the following lemma that the dual graph of a graph
manifold, that is, the graph on the set of facets in which two facets are
connected if they share a common ridge, is at least 2-connected.
Lemma 4.4.9. Let d ≥ 0, M be a d-graph manifold, and F̃ a facet of M.
Then for any two facets F =
6 F̃ and G =
6 F̃ there is a strong chain from F
to G that avoids F̃.
4.5
Connectivity of Graph Manifold Skeleta
Definition 4.5.1. Let M be a d-graph manifold and let −1 ≤ k ≤ ℓ ≤ d.
The (k, ℓ)-incidence graph Gk,ℓ (M) is the graph on the k-faces of M
in which two k-faces are connected by an edge if they are contained in a
common ℓ-face of M. We write Gk (M) for the (k, k + 1)-incidence graph
of M.
In this section we prove the following theorem.
Theorem 4.5.2. Let d ≥ −1 and −1 ≤ k ≤ d − 1. Let M be a weakly
normal d-graph manifold, and let Gk (M) be the (k, k + 1)-incidence graph
of M. Then the graph Gk (M) is
• (d + 1)-connected if k = d − 2, and
• (k + 2)(d − k)-connected, if −1 ≤ k ≤ d − 3 or k = d − 1.
We define the function m̃k (d) for −1 ≤ k ≤ d − 1 as
½
d + 1,
k =d−2
m̃k (d) :=
(k + 2)(d − k),
otherwise.
Thus, Theorem 4.5.2 states that the graph Gk (M) of a weakly normal
d-graph manifold is m̃k (d)-connected.
64
4.5. Connectivity of Graph Manifold Skeleta
The case k = d − 1 directly follows from Corollary 4.4.3: A d-graph
manifold has at least d + 2 facets, and Gd−1 (M) is the complete graph on
the set of facets of M. The case k = −1 is Theorem 4.4.7.
The proof of the other cases is split into the next three sections:
• We prove the special case k = d − 2 in Section 4.5.1.
• The other case is proved by induction on k. The base case of the
induction is Balinski’s theorem for graph manifolds. However, the
case k = 0 needs slightly different arguments than the case k ≥ 1,
and we start with this case in Section 4.5.2.
• In Section 4.5.3 we finish the induction for all 1 ≤ k ≤ d − 3.
4.5.1
The Case k = d − 2.
Lemma 4.5.3 (Barnette [8, Lemma 1]). Let d ≥ 1, and let
C = (F 0 , R1 , F 1 , . . . , Rn , F n )
be a strong chain of distinct facets in a d-graph manifold such that each
facet of C contains a fixed (d − 3)-face H. Then there is a strong chain of
distinct facets
C ′ = (F 0 , R1 , F 1 , . . . , Rn , F n , Rn+1 , F n+1 , . . . , Rn+k , F n+k = F 0 )
such that each facet of C ′ contains H. Such a strong chain is called a strong
cycle.
The proof of the following lemma is, at its core, identical to Athanasiadis’
proof in the polytope case [3].
Lemma 4.5.4. Let M be a d-graph manifold. Then the graph Gd−2 (M) is
(d + 1)-connected.
Proof. We can assume that d ≥ 1, as the other cases are trivial.
Let S be a subset of the set of (d − 2)-face of M of cardinality at most
d, and let F, G be two (d − 2) faces of M that are not in S.
A strong chain C = (F 0 , R1 , F 1 , . . . , Rn , F n ) from a facet F 0 that contains F to a facet F n that contains G naturally corresponds to a walk in
Gd−2 (M).
Suppose that for some i ∈ {1, . . . , n} the ridge Ri is a member of S.
Because of the cardinality of S and by Corollary 4.4.3, there is a (d−3)-face
H of Ri such that Ri is the only face in S that contains the face H. We
then construct a strong cycle of facets as in Lemma 4.5.3 around the face
H. This yields a strong chain of facets from F i−1 to F i that avoids the
ridge Ri and, by the choice of H, any element of S.
65
In the proof we have gotten a glimpse of the idea of the strong walks:
Lemma 4.5.3 essentially says that the strong components of the face figure
at a (d − 3)-face are 1-graph manifolds. Because the walk in Lemma 4.5.4
is a strong chain, we stay in one strong component of the face figure at a
(d − 3)-face at every forbidden face we have to avoid.
4.5.2
The Case k = 0.
Let M be a weakly normal graph manifold and let S be a subset of the
edges of M.
Let v be a vertex of M and Mv one of the strong components of M /v.
Define s(v, Mv ) to be the number of edges in S that correspond to vertices
in the component Mv .
Let e = uv be an edge of M, choose a strong component Mu of
M /u and a strong component Mv of M /v, and define t(e, Mu , Mv ) :=
s(u, Mu ) + s(v, Mv ).
Let (W, χ) be a strong walk with W = (v0 , e1 , v1 , . . . , en , vn ). As defined
in Section 4.4.2, let V 0 , . . . , V n be the strong components of the vertex
figures along W induced by the strong walk (W, χ). We then write s(i) for
s(vi , V i ) and t(i) for t(ei , V i−1 , V i ).
Definition 4.5.5 (Connecting walk, good walk). Let (W, χ) be a strong
walk in a weakly normal graph manifold M with
W = (v0 , e1 , v1 , . . . , en , vn ), and
C χ = (F 0 , R1 , F 1 , R2 , . . . , Rm , F m ).
If F1 , G1 ∈ M are faces of M with v0 ∈ F1 ⊆ F 0 and vn ∈ G1 ⊆ F m , we
say that the strong walk (W, χ) connects F1 and G1 .
We say that the strong walk (W, χ) is good (in the case k = 0), if it
satisfies the following two conditions:
(i) s(i) ≤ d − 1, for 0 ≤ i ≤ n and
(ii) t(i) ≤ d, for all 1 ≤ i ≤ n with ei ∈ S.
The following lemma shows that a good walk that connects two k-faces
(here, k = 0) yields the desired walk in Gk (M).
Lemma 4.5.6. Let d ≥ 3, M be a weakly normal d-graph manifold and S
a subset of the edges of cardinality less than 2d. Let F and G be edges of M
that are connected by a good walk. Then there exists a walk in G0 (M) \ S
that connects F and G.
66
W = (v0 , e1 , v2 , e2 , . . . , v5 )
F
v0
a2
b2
G
e4
e2
b4
v5
a4
Figure 4.3: The choice of edges along a good walk in M. In this example, the
edges e2 and e4 are the only edges of W that are members of S. For all other
edges we have ai = bi = ei (i = 1, 3, 5). Furthermore, a0 = F and b6 = G.
Proof. Let (W, χ) be a good walk that connects F and G with W =
(v0 , e1 , v1 , . . . , en , vn ). Let V 0 , . . . , V n and E 1 , . . . , E n be the components
of the vertex figures and edge figures along the walk (W, χ), respectively,
that are induced by the walk (W, χ). For every i ∈ {1, . . . , n}, the graph
manifold E i is a (d − 2)-graph manifold and therefore has at least d vertices,
by Corollary 4.4.3.
Let J ⊆ {1, . . . , n} be the set of indices of edges of W such that ej ∈ S
for every j ∈ J.
For every j ∈ J we choose two edges aj and bj in M in the following
way: By Definition 4.5.5 (ii), there is a vertex in E j that corresponds to a
1-face H in M such that the two edges of H that intersect ej are not in S.
Let bj be the edge of H that intersects ej in vj−1 , and let aj be the edge
of H that intersects ej in vj . The edges aj and bj are neighbors in G0 (M),
because they both lie in the 1-face H.
For every j ∈ {1, . . . , n} \ J we set aj = bj = ej . Finally, let a0 = F and
bn+1 = G.
For 0 ≤ i ≤ n, observe that ai and bi+1 are edges of M \ S which share
vi as a common endpoint; compare Figure 4.3.
Furthermore, for every i ∈ {0, . . . , n}, the vertices ai /vi and bi+1 /vi
both lie in V i . The graph manifold V i is of dimension d − 1 and therefore
d-connected. By Definition 4.5.5 (i), there are at most d − 1 vertices in V i
that correspond to elements in S. The graph G(V i ) minus these vertices is
therefore connected by Theorem 4.4.7. Thus, ai and bi+1 can be connected
by a walk in G0 (M) \ S. Therefore, F = a0 and G = bn+1 can also be
connected in G0 (M) \ S.
It remains to show that a good walk always exists. This is established
by the following lemma.
67
Lemma 4.5.7. Let d ≥ 3, M be a weakly normal d-graph manifold and S
a subset of the edges of M of cardinality less than m̃k (d) = 2d. For any
two edges F, G of M \ S there is a good walk that connects F and G.
Proof. We call a vertex v of M bad if it is an endpoint of at least (d + 2)/2
edges in S. Otherwise we call it good. Call an edge bad if it intersects at
least d + 1 edges in S, otherwise good. If (W, χ) is a strong walk in M and
W avoids bad vertices, then (W, χ) is a good walk. The number p of bad
vertices satisfies p⌈(d + 2)/2⌉ ≤ 2| S | ≤ 2(2d − 1). Since d ≥ 3 we have
p ≤ d, that is, there are at most d bad vertices.
If there are facets that contain all bad vertices let F̃ be one of them.
Otherwise, let F̃ be any facet.
Let F be a facet with F ∈ F and G a facet with G ∈ G and F =
6 F̃
and G =
6 F̃. Furthermore, let C = (F = F 0 , R1 , F 1 , . . . , Rn , F n = G) be
a strong chain that connects F and G and that avoids F̃. This exists by
Lemma 4.4.9. Then, for any good vertex v in F and any good vertex w in
G, there is a good walk from v to w.
Thus, it suffices to show that we can begin a good walk at one of the
endpoints of F and end a good walk at one of the endpoints of G.
Suppose F = ab and G = xy. Let
•
•
•
•
Ma be the strong component of M /a that contains F /a,
Mb be the strong component of M /b that contains F /b,
Mx be the strong component of M /x that contains G /x,
My be the strong component of M /y that contains G /y.
We distinguish in the following two cases.
Case (i). Consider first the edge F , and suppose that one of the endpoints of F , without loss of generality a, satisfies s(a, Ma ) ≥ d − 1. The set
S has at most 2d − 1 elements and F ∈
/ S, thus we must have s(b, Mb ) ≤
d − 2. Thus there is an edge HF that corresponds to a vertex in Mb that
does not belong to S. Simple counting shows that one of the endpoints of
HF is good and that HF is good.
If we assume that one of the endpoints of G, say x, satisfies s(x, Mx ) ≥
d − 1, we analogously derive the existence of an edge HG at y with the
desired properties.
Case (ii). Suppose that s(a, Ma ) ≤ d − 2 and s(b, Mb ) ≤ d − 2. Then
each of a and b is an endpoint of one edge of M that corresponds to a vertex
in Ma or Mb , respectively, that is not in S. We can further assume that
both a and b are bad. Suppose first that there are distinct vertices a′ and b′
of Ma and Mb that correspond to edges of M that connect a to a′ and b to
b′ . Counting shows that at least one of a′ , b′ is good. In this case, we have
68
the existence of a good edge HF that is incident to F , that corresponds to
a vertex in either Ma or Mb , depending which of the vertices a′ and b′ is
good, and that has a good endpoint.
Otherwise, both a and b must be connected to a vertex c of M by edges
that correspond to vertices in Ma and Mb , respectively, and that are not in
S. Furthermore, we have s(a, Ma ) = s(b, Mb ) = d − 2. Then s(c, Mc ) ≤ 1
for any strong component Mc of M /c and c is a good vertex. In this case,
let HF be the edge from a to c. Then HF is good.
Similarly, if we assume that s(x, Mx ) ≤ d − 2 and s(y, My ) ≤ d − 2, we
derive the existence of an edge HG with the desired properties.
Let q1 ∈ {a, b} be the common vertex of F and HF , let q2 ∈ {x, y}
be the common vertex of G and HG , let Mq1 be the strong component of
M /q1 that contains the facet F /q1 , and let Mq2 be the strong component
of M /q2 that contains the facet F /q2 .
Let F ′′ be a facet with HF ∈ F ′′ and F ′′ /q1 ∈ Mq1 and find a strong
chain C 1 from F to F ′′ that avoids F̃ such that every facet of C 1 contains
q1 . Let G ′′ be a facet with HG ∈ G ′′ and G ′′ /q2 ∈ Mq2 and find a strong
chain C 2 from G to G ′′ that avoids F̃ such that every facet of C 2 contains
q2 . Both C 1 and C 2 can be chosen to avoid F̃ by Lemma 4.4.9.
−1
−1
The chain C := C 1 C −1
is the
1 C C 2 C 2 (where for a chain C̃ the chain C̃
strong chain on the same set of facets traversed in the opposite direction)
is a strong chain of M. It supports a good walk (W, χ̃) that connects F
and G.
4.5.3
The Case k ≥ 1.
Let M be a weakly normal graph manifold and let k ≥ 1. Let S be a subset
of the k-faces of M of cardinality less than m̃k (d) = (k + 2)(d − k).
Let (W, χ) be a strong walk with W = (v0 , e1 , v1 , . . . , en , vn ), and let
V0 , . . . , V n , E 1 , . . . , E n be the strong components of the vertex figures and
edge figures, respectively, along W that are induced by (W, χ).
For i ∈ {0, . . . , n}, define s(i) to be the number of k-faces in S that
correspond to (k − 1)-faces in V i .
For i ∈ {1, . . . , n}, define t(i) to be the number of k-faces in S that
correspond to (k − 2)-faces in E i .
Definition 4.5.8 (Good walk). We say that a strong walk (W, χ) is good
if it satisfies the following two conditions:
(i) s(i) < m̃k−1 (d − 1) for all i ∈ {0, . . . , n}, and
¡¢
(ii) t(i) < kd for all i ∈ {1, . . . , n}.
69
As in the case k = 0, the existence of a good walk establishes the
existence of the desired walk in Gk (M).
Lemma 4.5.9. Let d ≥ 4, k ≥ 1, M be a weakly normal d-graph manifold
and S a subset of the k-faces of M of cardinality less than m̃k (d) = (k +
2)(d − k).
Let F and G be two k-faces of M and suppose that (W, χ) is a good
walk that connects F and G. Then there exists a walk in Gk (M) \ S that
connects F and G.
Proof. Let (W, χ) be a good walk that connects F and G with
W = (v0 , e1 , v1 , . . . , en , vn )
C χ = (F 0 , R1 , F 1 , . . . , Rm , F m ).
Set G0 := F and Gn+1 := G. Let V i and E i be the strong components
of M /vi and M /ei , respectively, that are induced by (W, χ). For every i
choose a k-face Gi in M \ S with Gi /ei ∈ E i . One sees that such a face
exists, using condition (ii) of Definition 4.5.8 and Corollary 4.4.3.
Now, vi is a vertex of Gi and Gi+1 . Furthermore, Gi /vi and Gi+1 /vi
both lie in V i . By induction and by Definition 4.5.8 (i), there is a walk in
Gk−1 (V i ) that avoids the (k − 1)-faces of V i that correspond to faces in S
and that connects Gi /vi and Gi+1 /vi . Thus, also F = G0 and G = Gn+1
can be connected by a walk in Gk (M).
Before we can show that a good walk always exists, we need one more
lemma. This lemma is the combinatorial analogon to a simple geometric
statement: If any (k + 2) points of a set in Rd lie in a k-flat, that is, are
affinely dependent, then the whole set lies in a k-flat.
Lemma 4.5.10. Let U ⊆ V(M) be a subset of the vertices
¡ U ¢ of a graph
there is a
manifold M and k ≥ 2. Suppose that for every U ′ ∈ k+2
(k−1)-face that contains U . Then there is a single (k−1)-face that contains
U.
Proof. Suppose there is a set U ′ of cardinality k + 1 such that the smallest
face F that contains U ′ is of dimension k − 1. By assumption, U ′ ∪ {v} is
contained in a (k − 1) face F ′ and necessarily F ′ = F , by the intersection
property (otherwise F ′ ∩ F is a smaller face than F that contains U ′ ).
If no such set as U ′ exists, then the smallest face that contains a set
U ′′ ⊆ U of cardinality k + 1 is of dimension less than k − 1 for any set U ′′ .
Thus, by induction the statement follows, where for k = 2 only the first
case applies.
70
Finally, we show that a good walk always exist.
Lemma 4.5.11. Let d ≥ 4, k ≥ 1, and let M be a weakly normal d-graph
manifold and S a subset of the k-faces of M of cardinality less than m̃k (d) =
(k + 2)(d − k). For any two k-faces F, G of M \ S there is a good walk that
connects F and G.
Proof. To prove the lemma we distinguish the cases k = 1 and k ≥ 2.
Case k = 1. The set S is a subset of the 1-faces of M, m̃k (d) = 3d − 3,
and m̃k−1 (d − 1) = 2d − 2. Call an edge or vertex of M bad if it is contained
in at least d or 2d − 2 elements of S, respectively.
We show that
(a) there are at most two bad edges,
(b) if v, w are two distinct bad vertices, then u and v are connected by a
bad edge, and
(c) if v is a bad vertex and e is a bad edge, then v is a vertex of e.
Suppose there are three bad edges e1 , e2 , e3 . Then for i ∈ {1, 2, 3} the
edge ei is incident to at least d edges. Any two of these three edges share
at most one 1-face of S. Thus,
| S | ≥ 3d − 3.
This contradicts that | S | < m1 (d) = 3d − 3 and (a) is proved.
If v and w are distinct bad vertices, then v and w are contained in at
least 2d − 2 elements of S each. Since | S | < 3d − 3 they share at least d
of them, which, by the intersection property, intersect in an edge e. This
shows (b).
If v is a bad vertex and e is a bad edge, then there are at least two
elements of S that contain both v and e and (c) follows.
It follows that there is a set U of at most two vertices such that G(M)\U
contains no bad vertex and no bad edge. Thus G(M) \ U contains a good
walk that connects F and G: Simply choose any strong chain C from a
facet that contains F to a facet that contains G. Then there is a walk in
the boundary that visits the facets of C consecutively and avoids U . This
is possible since d ≥ 4.¡ ¢
Case k ≥ 2. Since kd ≥ (k + 1)(d − k) = m̃k−1 (d − 1) for d ≥ k + 3 ≥ 5,
the condition (ii) in Definition 4.5.8 follows from (i).
Let U be the set of bad vertices, that is, those that are contained in at
least m̃k−1 (d − 1) elements of S. Let U ′ ⊆ U be a set of size k + 2. Since S
has less than m̃k (d) = (k + 2)(d − k) elements and each u ∈ U is contained
in at least (k + 1)(d − k) of them, there are at least k + 1 elements in S
71
that contain U ′ . That is, for every subset of U of size k + 2 the vertices of
this subset all lie in a common (k − 1)-face of M. By Lemma 4.5.10, there
is a single (k − 1)-face H that contains U .
Let F ′ ∈ M be a facet that contains the face H. Choose facets F and
G with F ∈ F and G ∈ G with F =
6 F ′ and G =
6 F ′ , and let
C = (F = F 0 , R1 , F 1 , . . . , Rm , F m = G)
be a strong chain between F and G that avoids F ′ —this exists, as the dual
graph is at least 2-connected by Lemma 4.4.9.
Let v ∈ F and w ∈ G with v, w ∈
/ U . These exist because U is contained
in a (k − 1)-face of M. The intersection of F ′ with every Ri of the strong
chain C is at most a (d − 3)-graph manifold, thus for every Ri there is a
vertex not in U . Also, the graph of every F i minus F ′ ∩ F i is connected,
by Lemma 4.4.8. Thus, there is a walk W from v to w that visits the facets
of C consecutively and that avoids the vertices in U .
This also concludes the proof of Theorem 4.5.2.
4.6
Proof of Athanasiadis’ Conjecture
Theorem 4.6.1. Let M be a weakly normal (d + 1)-graph complex, and let
Gk (M) be the (k, k + 1)-incidence graph, for −1 ≤ k ≤ d − 2. Then
½
d + 1,
k =d−2
κ(Gk (M)) ≥
(k + 2)(d − k),
otherwise
Proof. Recall that
m̃k (d) :=
½
d + 1,
(k + 2)(d − k),
k =d−2
otherwise,
and let S be a set of k-faces of cardinality less than m̃k (d).
The d-faces of M are weakly normal d-graph manifolds. Let F, G be
two k-faces of M, let F and G be two facets of M such that F ⊆ F and
G ⊆ G. Choose a strong chain C = (F = F 0 , R1 , F 1 , . . . , Rn , F n = G) from
F to G.
Then G k (F i ) is at least m̃k (d)-connected, for every
¡d+1¢ i ∈ {0, . . . , n} by
Theorem 4.5.2. Furthermore, every Ri has at least k+2 faces of dimension
k. Since we have
µ
¶
d+1
≥ (k + 2)(d − k), for k ≤ d − 3,
k+2
there is at least one k-dimensional face in Ri that is not in S.
Thus we can connect F and G by a walk in Gk (M).
72
4.6. Proof of Athanasiadis’ Conjecture
As was remarked in Section 4.3, because of technical difficulties with
vertex figures the methods of this chapter do not generalize in an obvious
way to the whole class of graph manifolds.
Conjecture 4.6.2. Let M be a (d + 1)-graph complex, and let Gk (M) be
the (k, k + 1)-incidence graph, for −1 ≤ k ≤ d − 2. Then
½
d + 1,
k =d−2
κ(Gk (M)) ≥
(k + 2)(d − k),
otherwise
Conjecture 4.6.2 is probably as general as one can get. Naatz [83] defines
face structures (not to be confused with our face structures) that generalize
Barnette’s graph manifolds and proves Balinski’s theorem for them. His
definition has the advantage that every d-connected graph supports a face
structure of rank d [83, Proposition 3.2.3]. This also means that his face
structures are far away from resembling any geometric situation. For example, there are face structures in which the faces of rank one can have
arbitrarily many vertices.
Of course, the choice of graph manifold as the structure in which Conjecture 4.6.2 is stated is rather arbitrary. It would probably be more natural
(and the definitions less convoluted) to state this conjecture for graded
relatively complemented lattices with some additional properties (strongly
connectedness, pseudo-manifold property); compare with the concept of
abstract polytopes [51] [62]. Solely for historical reasons, I prefer graph
manifolds to other terminology.
Lemma 4.6.3. Let C be a d-dimensional Cohen-Macaulay regular cell complex with intersection property. Then the graphs of the cells of C form a
normal d-graph complex.
Proof. Let M be the d-graph complex formed by the graphs of the cells of
the cell complex C.
Then M satisfies (1), as a “cell of a cell of C is a cell of C”; it satisfies (3)
by the intersection property of C; the complex C is strongly connected and
pure by Cohen-Macauliness [3], that is, the complex M satisfies (4) and (2);
finally, every face F of M satisfies the pseudomanifold property (5) as the
boundary of the corresponding cell of C is a cell decomposition of a sphere;
compare [23, p. 204]. Furthermore, because every vertex figure of a manifold
is a homology sphere, see Cairns [34, Chapter 7] or Munkres [82, §63], and
thus strongly connected, the complex is normal.
This finally proves Athanasiadis’ conjecure, Conjecture 4.2.1.
73
Corollary 4.6.4 (Conjecture 6.2 [3]). For any d-dimensional CohenMacaulay regular cell complex C with intersection property, the graph Gk (C)
is
• (k + 1)(d − k)-connected if 0 ≤ k ≤ d − 3,
• d-connected if k = d − 2.
Proof. This follows from Lemma 4.6.3 and Theorem 4.6.1.
74
Chapter 5
Linkages in Polytope Graphs
Linkages are a very important concept in graph theory. They play a major
role in the theory of minors, and they are in a very strong sense related
to connectivity; see Bollobás & Thomason [28], Kostochka [65], Larman
& Mani [66], Robertson & Seymour [95], Thomas & Wollan [113], and
Thomason [114].
In this chapter, we consider linkages in graphs of polytopes. These were
first studied by Larman & Mani [66] and later by Gallivan [44] [45].
The Handbook of Discrete and Computational Geometry states the following question by Larman & Mani [66] as a problem:
Question 5.0.1 (Larman & Mani [66], Kalai [63, Problem 20.2.6]).
Let G be the graph of a d-polytope and k = ⌊d/2⌋. Is it true that for every
two disjoint sequences (s1 , . . . , sk ) and (t1 , . . . , tk ) of vertices of G there are
k vertex-disjoint paths connecting si to ti , i = 1, . . . , k?
This question asks, rephrased in customary graph theory language,
whether the graph of every d-polytope is ⌊d/2⌋-linked.
We can answer in the affirmative in dimensions d ≤ 5: This is trivial
in dimensions d = 0, 1, 2. In dimension d = 3 a polytope is 2-linked if and
only if it is simplicial and otherwise 1-linked; see Figures 5.1(a) and 5.1(b)
for illustrations of these cases. Every 4-polytope and every 5-polytope is
2-linked—this follows from the characterization of 2-linked graphs by Seymour [100], Shiloach [102], and Thomassen [115]; results by Jung [58]; or a
geometric argument similar to those employed in some proofs of Balinski’s
75
5. Linkages in Polytope Graphs
t2
t1
t2
t1
s1
s2
s1
s2
(a) Simplicial 3-polytopes are 2-linked.
(b) Every path from s1 to t1 disconnects
s2 and t2 .
Figure 5.1: Linkages in simplicial polytopes and 3-dimensional polytopes.
theorem, as in [118, Theorem 3.14].
In higher dimensions the situation is quite different: Question 5.0.1 has
a negative answer in dimensions 8, 10, and d ≥ 12. Even when k is chosen
as ⌊2(d + 4)/5⌋ (which is strictly smaller than ⌊d/2⌋ for all d ≥ 22), dpolytopes are not necessarily k-linked. Indeed, polytopes with this property
were already discovered in the 1970s by Gallivan [44, Theorem 7, p.46] and
later published by McMullen [78] and Gallivan [45].
We denote by k(d) the largest integer such that every d-polytope is k(d)linked. Gallivan’s examples show that k(d) ≤ ⌊(2d + 3)/5⌋, and Larman &
Mani [66] have proven a lower bound of ⌊(d + 1)/3⌋. We improve this lower
bound marginally to ⌊(d + 2)/3⌋ in Section 5.3. Of course, this improvement
is irrelevant for asymptotic questions, but it implies exact values for k(d)
in dimensions 7, 10, and 13.
In the special case of simplicial polytopes, a precise answer was given
by Larman & Mani [66]. They have shown that every simplicial d-polytope
is ⌊(d + 1)/2⌋-linked. The stacked polytopes show that this bound cannot
be improved.
A closer look at Gallivan’s examples made it apparent that minimal
linkedness of d-polytopes on f0 = d + γ + 1 vertices does depend on γ,
at least if γ is small. In that respect, linkedness behaves differently than
76
5.1. Linkages
connectivity in the polytope case.
We therefore introduce a new parameter k(d, γ) that measures minimal
linkedness of d-polytopes on d + γ + 1 vertices. We determine k(d, γ) for
polytopes on at most (6d + 7)/5 vertices in Section 5.4 and analyze the
combinatorial types of polytopes with linkedness exactly k(d, γ).
Among the combinatorial types that meet the lower bound, Gallivan’s
polytopes are in some sense the canonical ones, in some cases even unique: If
f0 is odd there is only one combinatorial type with linkedness k(d, γ) among
all polytopes on f0 vertices. This type is given by an iterated pyramid over
a join of quadrilaterals. Because of the special combinatorial structure of
these polytopes they are even projectively unique.
We complement this result by showing that, if f0 is even, there are
“many” combinatorial types of polytopes with minimal linkedness k(d, γ).
5.1
Linkages
Definition 5.1.1 (k-linked, linkage). Let G = (V, E) be a graph.
The graph G is k-linked if |V | ≥ 2k and for every choice of 2k distinct
vertices s1 , . . . , sk , t1 , . . . , tk there exist k vertex-disjoint paths L1 , . . . , Lk
such that Li joins si and ti for i = 1, . . . , k.
If we write s = (s1 , . . . , sk ) and t = (t1 , . . . , tk ), we call the paths
L1 , . . . , LK an (s, t)-linkage.
We denote by k(G) the largest integer k such that G is k-linked.
Definition 5.1.2 (Linkage parameters). We say that a polytope P is
k-linked if the graph G(P ) is k-linked and define the following parameters
for general polytopes:
k(P ) := k(G(P )),
k(d, γ) := min{k(P ) : P ∈ P γd },
k(d) := min k(d, γ).
γ
For simplicial polytopes we define:
kS (d, γ) := min{k(P ) : P ∈ S γd },
kS (d) := min kS (d, γ).
γ
We call k(d) the minimal linkedness of d-polytopes and kS (d) the minimal
linkedness of simplicial d-polytopes.
77
In this chapter we use the following convenient notation for subpaths of
a path L = v0 v1 . . . vk (found in [38]),
Lvi := v0 . . . vi ,
vj L := vj . . . vk ,
vj Lvi := vj . . . vi ,
where j ≤ i for i, j ∈ {0, . . . , k}.
5.2
Simplicial Polytopes and 3-Polytopes
We determine the minimal linkedness of simplicial polytopes by reproving
a theorem by Larman & Mani [66].
From this result, we derive an exact criterion for the linkedness of 3polytopes.
5.2.1
Linkedness of Simplicial Polytopes
It is almost trivial that a simplicial d-polytope is ⌊d/2⌋-linked; compare
Lemma 5.4.1. To get the tight bound of ⌊(d + 1)/2⌋ proved by Larman &
Mani [66] we have to work harder.
Lemma 5.2.1. Let P be a simplicial d-polytope and v be a vertex of P .
Then v is connected to every vertex of linkB(P ) (v) by an edge.
Proof. The set of vertices in the link consists of precisely the vertices in
facets incident to v, the vertex v excluded. Since P is simplicial, all these
facets are simplices and thus have complete graphs.
The following theorem was shown by Larman & Mani [66]. Their proof
is similar to ours. See Figure 5.2 for an illustration of the 4-dimensional
case.
Theorem 5.2.2 (Larman & Mani [66]). Let d ≥ 2, γ ≥ 0. Then
º
¹
d+1
.
kS (d, γ) = kS (d) =
2
Proof. We begin by proving the lower bound. Clearly, if d = 2 then
kS (d, γ) = kS (d) = 1.
If d = 3, let s = (s1 , t1 ) and t = (s2 , t2 ), where s1 , s2 , t1 , t2 are distinct
vertices of a simplicial 3-polytope P .
78
5.2. Simplicial Polytopes and 3-Polytopes
t1
t′1
s′2
t2
t′′1
s′1 = s′′1
s2
s1
Figure 5.2: The link Q = linkB(P ) (t2 ) in a 4-polytope P . The path from s2 to
Q extends to a path from s2 to t2 . The paths from s1 and t1 to Q can be joined
in R = linkQ (s′2 ).
The graph G(P ) is 3-connected by Balinski’s theorem. Thus by Menger’s
theorem there exist disjoint paths S1 , T1 , S2 from {s1 , t1 , s2 } to t2 , such that
S1 ∩linkB(P ) (t2 ) = {s′1 },
T1 ∩linkB(P ) (t2 ) = {t′1 },
S2 ∩linkB(P ) (t2 ) = {s′2 }.
These conditions can be assumed by Lemma 5.2.1.
Since Q := linkB(P ) (t2 ) is combinatorially isomorphic to the boundary
complex of a 2-polytope and thus its graph is 2-connected, there is a path
S1′ in Q from s′1 to t′1 that avoids s′2 . Consequently, the paths
L1 := (S1 s′1 )S1′ (t′1 T1 ) and L2 := S2
are an (s, t)-linkage in P .
Let d ≥ 4, and let P be a simplicial d-polytope. Let k = ⌊(d + 1)/2⌋,
and let s = (s1 , . . . , sk ) and t = (t1 , . . . , tk ), where s1 , . . . , sk , t1 , . . . , tk are
distinct vertices of P . We have to construct an (s, t)-linkage in G(P ).
Consider Q := linkB(P ) (tk ), the link of P at tk . Then Q is combinatorially isomorphic to the boundary complex of a vertex figure at tk , that is,
to the boundary complex of a simplicial (d − 1)-polytope.
Since d ≥ 2k − 1 and by Balinski’s theorem, there exist 2k − 1 vertexdisjoint paths S1 , . . . , Sk and T1 , . . . , Tk−1 that connect the vertices s1 , . . . , sk
and the vertices t1 , . . . , tk−1 to tk .
Since P is simplicial we can assume that each of the paths hits Q exactly
once by Lemma 5.2.1. Let s′1 , . . . , s′k and t′1 , . . . , t′k−1 be the intersection
79
vertices, that is,
{s′i } = Si ∩ Q and {t′j } = Tj ∩ Q
for each i = 1, . . . , k and j = 1, . . . , k − 1.
Let R := linkQ (s′k ). Since G(Q) is at least (2k − 2)-connected, we
′
′
find vertex-disjoint paths S1′ , . . . , Sk−1
and T1′ , . . . , Tk−1
in Q that connect
′
′
′
′
′
s1 , . . . , sk−1 and t1 , . . . , tk−1 to sk . We can assume that these paths hit
R exactly once with intersection vertices s′′1 , . . . , s′′k−1 and t′′1 , . . . , t′′k−1 by
Lemma 5.2.1.
The polytope R is simplicial and (d − 2)-dimensional. Therefore it is
⌋-linked by induction, and we have k − 1 = ⌊ d−1
⌋. Let L′′1 , . . . , L′′k−1
⌊ d−1
2
2
be an (s′′ , t′′ )-linkage for s′′ = (s′′1 , . . . , s′′k−1 ) and t′′ = (t′′1 , . . . , t′′k−1 ) in R.
Then
½
(Si s′i )(Si′ s′′i )L′′i (t′′i Ti′ )(t′i Ti ),
1≤i≤k−1
Li =
Sk ,
i=k
is an (s, t)-linkage in P .
To prove the upper bound observe that any stacked polytope on d+γ +1
vertices has a separating set of vertices of size d ≤ 2⌊ d+1
⌋.
2
Larman & Mani [66] have proven Theorem 5.2.2 in more generality.
They define a d-simplicial graph as follows: A 2-simplicial graph is a cycle.
For d ≥ 3, a graph G is d-simplicial if it is connected and the graph induced
by the neighbors N (v) of any vertex v contains a (d − 1)-simplicial ¥graph
¦
with N (v) as vertices. They have shown that a d-simplicial graph is d+1
2
linked. It is easy to see that the graph of a simplicial d-polytope is dsimplicial (induction on d by taking vertex links).
The proof given above can be adapted to this situation, based on the
fact that a d-simplicial graph is d-connected. This was shown in [66].
5.2.2
Linkedness of 3-Polytopes
In order to write down an exact statement for linkedness of 3-polytopes we
determine the parameter
K(d, γ) := max{k(P ) : P ∈ P γd },
that is, the maximal linkedness of a d-polytope on d + γ + 1 vertices.
It is easily observed that

1
,
d = 1, 2

2
,
d=3
K(d, γ) =
 ¥ d+γ+1 ¦
,
d
≥ 4.
2
80
5.3. Minimal Linkedness of Polytopes
This is trivial for d = 1, 2, while for d = 3 the statement follows from
planarity. For d ≥ 4 there are polytopes with graph Kd+γ+1 for any γ ≥ 0—
take the d-dimensional cyclic polytope on d + γ + 1 vertices, for instance.
As a consequence, we can precisely describe the linkedness of a given
d-polytope when d ≤ 3 in the following corollary to Theorem 5.2.2.
Corollary 5.2.3. If P is a 1- or 2-polytope, then k(P ) = 1.
If P is a 3-polytope, then k(P ) ≤ 2 and k(P ) = 2 if and only if P is
simplicial.
5.3
Minimal Linkedness of Polytopes
In this section we provide lower and upper bounds on k(d) for general
polytopes in arbitrary dimension d. We prove an upper bound on k(d, γ)
that is independent of γ.
5.3.1
Lower Bound on Minimal Linkedness
There are general results in graph theory that provide a link1 between
connectivity and linkages.
A k-linked graph is at least (2k −1)-connected, because a k-linked graph
cannot have a separating set of size 2k − 2.
The proof that a highly connected graph is also highly linked is nontrivial. The most recent result of that type is by Thomas & Wollan [113], who
have shown that every 10k-connected graph is k-linked.
Balinski’s theorem thus implies that every d-polytopal graph is at least
⌊d/10⌋-linked. The bound of ⌊d/10⌋ can be greatly improved by looking
at large complete topological minors of polytope graphs. These exist by
Corollary 2.1.5.
In fact, Larman & Mani [66] have shown that every 2k-connected graph
that contains a K3k as a topological minor is k-linked. Robertson & Seymour [95] have proven the much stronger statement that one may replace
“topological minor” by “minor” in the previous statement.
Consequently, every d-polytope is ⌊(d + 1)/3⌋-linked. However, already
in dimension 4 this bound is not tight. It is easy to see by a geometric argument that every 4-polytope is 2-linked (as was remarked in the introduction, this also follows from the characterization of 2-linked graphs by Seymour [100], Shiloach [102], and Thomassen [115], or results by Jung [58]).
1
or : a connection
81
We improve Larman & Mani’s bound slightly by considering rooted
topological minors of d-polytopes. The proof of the following lemma is a
variation of an argument by Diestel [38, pp. 70–71].
Lemma 5.3.1. Let G = (V, E) be a 2k-connected graph. Suppose that for
every vertex v of G the graph G contains a subdivision of K3k−1 rooted at
v. Then G is k-linked.
Proof. Let s = (s1 , . . . , sk ) and t = (t1 , . . . , tk ), where s1 , . . . , sk , t1 , . . . tk
are distinct vertices of G. Let K be a subdivision of K3k−1 rooted at vertex
tk with principal vertices U := U ′ ∪ {tk }, where U ′ ⊆ N (tk ).
Since G \ {tk } is (2k − 1)-connected there exist 2k − 1 disjoint paths
S1 , . . . , Sk and T1 , . . . , Tk−1 in G that avoid tk such that Si joins si to U ′ ,
for i = 1, . . . , k, and Ti joins ti to U ′ , for i = 1, . . . , k − 1. Moreover, we
assume that the paths have been chosen such that they do not have interior
vertices in U ′ (and thus also not in U ) and that their total number of edges
outside of E(K) is minimal.
Let W = {v1 , . . . , vk , w1 , . . . , wk−1 } be the vertices of these paths in U ′ ,
where vi is in Si and wi is in Ti . We then have a partition of U into sets
{tk }, W and W ′ := U ′ \ W with |W ′ | = k − 1. Let u1 , . . . , uk−1 be the
vertices in W ′ ⊆ U . We call these vertices free.
Since the path Sk joins sk to a neighbor of tk the path Lk := Sk Tk joins
sk and tk , where Tk is the path that consists of the single edge from the
vertex sk to the vertex tk .
Now fix some i ∈ {1, . . . , k − 1} and let Mi be the path in K from the
free vertex ui to vi and Ni be the path in K from ui to wi . Since the paths
S1 , . . . , Sk , T1 , . . . , Tk−1 were chosen minimal with respect to their number
of edges outside of K and ui is a free vertex, the paths Sj are disjoint from
Mi for j 6= i, and they are disjoint from Ni for all j = 1, . . . , k. Similarly,
the paths Tj are disjoint from Ni for j 6= i, and they are disjoint from Mi
for all j = 1, . . . , k − 1. Hence we can join vi to wi via the free vertex ui .
Denote by s′i the intersection vertex of Si and Mi that is closest to ui ,
and by t′i the intersection vertex of Ti and Ni that is closest to ui . We then
get pairwise disjoint paths
½
1≤i≤k−1
(Si s′i )(s′i Mi )(Ni t′i )(t′i Ti ),
Li =
Sk Tk ,
i=k
such that Li joins si and ti , that is, an (s, t)-linkage.
See Figure 5.3 for an illustration of the proof.
Theorem 5.3.2. Every d-polytope is ⌊(d + 2)/3⌋-linked.
82
5.3. Minimal Linkedness of Polytopes
s1
v1
w1
tk
W
t1
vk
u1
W′
sk
Figure 5.3: Illustration of the proof of Lemma 5.3.1 with k = 2.
Proof. Let P be a d-polytope. We set k := ⌊(d + 2)/3⌋. For d ≥ 2, we then
have d ≥ 2k and d + 1 ≥ 3k − 1.
Therefore, by Corollary 2.1.5, the graph G(P ) contains, at every vertex
v, a K3k−1 subdivision rooted at v. By Balinski’s theorem, G(P ) is at least
2k-connected. Lemma 5.3.1 implies that the graph of P is k-linked.
5.3.2
Upper Bound on Minimal Linkedness
Theorem 5.3.3. Let d ≥ 2 and γ ≥ 1. Then the minimal linkedness of
d-polytopes on d + γ + 1 vertices satisfies
k(d, γ) ≤ ⌊d/2⌋ .
Proof. For d = 2 the assertion clearly is true.
Let d ≥ 3 and γ ≥ 1. To prove the statement we have to construct a
d-polytope on d + γ + 1 vertices with k(P ) ≤ ⌊d/2⌋.
Let Q be a 3-polytope on 4 + γ vertices that has a square facet. For
instance, for γ = 1 take the pyramid over a square and for γ > 1 stack this
pyramid γ − 1 times over triangular facets.
Let P := pyrd−3 (Q), the (d − 3)-fold pyramid over Q. Then P is a
d-polytope and has d + γ + 1 vertices.
We claim that P is not (⌊d/2⌋ + 1)-linked. To see this let s1 , t1 , s2 , t2 be
the vertices of a square facet of Q (in that order around the facet). Then,
by planarity, these cannot be linked in G(Q).
Additionally, with m = ⌊(d − 3)/2⌋ there are exactly 2m vertices in
V(P ) \ V(Q) if d is odd and exactly 2m + 1 if d is even. We choose distinct
83
vertices s3 , . . . , sm+2 , t3 , . . . , tm+2 arbitrarily from the set V(P ) \ V(Q) and,
if d is even, we let sm+3 be the last vertex left in V(P ) \ V(Q) and choose
tm+3 arbitrarily from V(Q) \ {s1 , s2 , t1 , t2 }.
This set of ⌊d/2⌋ + 1 pairs of vertices cannot be linked in P . Therefore
k(P ) ≤ ⌊d/2⌋.
In the special case γ = 0 we trivially have k(d, γ) = ⌊(d + 1)/2⌋, as the
d-simplex is ⌊(d + 1)/2⌋-linked.
Theorem 5.3.3 implies that k(d) ≤ ⌊d/2⌋. We improve this bound significantly in the next section.
5.4
Linkages in Polytopes with Few Vertices
We now study linkedness of d-polytopes that have only few vertices more
than the minimum of d + 1. One remark on the usage of the term “few
vertices” is in order: It is used here in a slightly different sense than usual.
It does not mean that γ = f0 − d − 1 is considered as a constant. However,
γ will not be larger than d in this section.
For γ ≤ (d+2)/5, we precisely determine the value of k(d, γ) and analyze
polytopes that attain the value of k(d, γ).
5.4.1
Lower Bound for Polytopes with Few Vertices
Linkedness of a graph is a local property in the following sense: If a graph
is highly connected, then a k-linked subgraph ensures k-linkedness for the
whole graph. The precise statement is the following lemma.
Lemma 5.4.1. Let G = (V, E) be a 2k-connected graph and G′ a subgraph
of G that is k-linked. Then G is k-linked.
Proof. Let s = (s1 , . . . , sk ) and t = (t1 , . . . , tk ), where s1 , . . . , sk , t1 , . . . , tk
are distinct vertices in G.
Since G is 2k-connected, there exist 2k vertex disjoint paths S1 , . . . , Sk
and T1 , . . . , Tk such that Si connects si to G′ and Ti connects ti to G′ . We
choose the paths such that each contains only one vertex from G′ . Let
{s′i } = G′ ∩ Si and {t′i } = G′ ∩ Ti , for i = 1, . . . , k.
Since G′ is k-linked there exists an (s′ , t′ )-linkage L′1 , . . . , L′k in G′ for
s′ = (s′1 , . . . , s′k ) and t′ = (t′1 , . . . , t′k ).
Then
Li = Si L′i Ti ,
1≤i≤k
is an (s, t)-linkage in G.
84
5.4. Linkages in Polytopes with Few Vertices
We obtain a lower bound on linkedness of polytopes with few vertices
by finding a highly-linked subgraph in the graph of P . This highly-linked
subgraph is a complete subgraph: the graph of a simplex face of high dimension.
Lemma 5.4.2 (see Marcus [72], Kalai [62]). Let P be a d-polytope on
d + γ + 1 vertices with d ≥ γ. Then P has a (d − γ)-face that is a simplex.
Proof. The statement is true for every 2-polytope on 3 + γ vertices, γ ≥ 0.
Let P be a d-polytope, d ≥ 3. Choose a facet F , which is of dimension
d′ = d − 1. Suppose that F has d′ + γ ′ + 1 vertices, where 0 ≤ γ ′ ≤ γ.
By induction, F has a simplex face S of dimension dim(S) = d′ − γ ′ =
d − 1 − γ ′ = d − (γ ′ + 1). If γ ≥ γ ′ + 1, then dim(S) ≥ d − γ and we are
done.
If γ = γ ′ , then V(P ) \ V(F ) = {v} and P = pyrv (F ), that is, P is a
pyramid over F with apex v. Hence pyrv (S) is a face of P and a simplex
of dimension dim S + 1 = d − γ.
Lemma 5.4.2 also follows from the Blumenthal–Robinson Theorem [26],
see Theorem 10.2.1, by considering Gale diagrams: A Gale diagram of a
polytope on f0 = d + γ + 1 vertices must contain a positive basis of size n
where γ + 1 ≤ n ≤ 2γ. This positive basis corresponds to a complement of
a simplex face of size f0 − n ≥ d − γ + 1.
For “large” γ, the bound given in Lemma 5.4.2 can be slightly improved;
see Marcus [72].
Lemma 5.4.3. Let d ≥ 2, and d ≥ γ ≥ 0. Then
¹
º
d−γ+1
k(d, γ) ≥
.
2
Proof. In the special case γ = 0 the assertion is trivially true.
For d ≥ 2, γ ≥ 1 it follows from Lemma 5.4.1 and Lemma 5.4.2, since
2⌊(d − γ + 1)/2⌋ ≤ d − γ + 1 ≤ d, and the graph of a d-polytope is at least
d-connected, by Balinski’s theorem.
5.4.2
Upper Bound for Polytopes with Few Vertices
To prove a good upper bound on the number k(d, γ) we have to find a
polytope P on d + γ + 1 vertices with small k(P ). For γ ≤ (d + 2)/5 the
lower bound of Lemma 5.4.3 can be attained.
The examples we describe here were first discovered in this context by
Gallivan [44] [45]. He constructed them using Gale diagrams.
85
Definition 5.4.4. For integers n, m ≥ 0 and j1 , k1 , . . . , jm , km ≥ 1 define
G (n, j1 , k1 , . . . , jm , km ) := ∆n−1 ∗ (∆j1 ⊕ ∆k1 ) ∗ · · · ∗ (∆jm ⊕ ∆km )
and
· · ∗ ¤} .
G (n, m) := G(n, 1, . . . , 1) = ∆n−1 ∗ |¤ ∗ ·{z
| {z }
m times
2m times
The parameters d and γ for G (n, m) can be determined by observing
that G (n, m) has 4m + n vertices, so d + γ + 1 = 4m + n, and dimension
dim(G (n, m)) = n − 1 + 3m. Therefore we have
d = n − 1 + 3m, and
(5.1)
γ = m.
(5.2)
We consider the complement graph G(G (n, m)) of G(G (n, m)) to examine the linkedness of the polytopes G (n, m). It is easy to see that the
graph of the join of two polytopes corresponds to the join of the graphs of
the polytopes. Thus, the complement graph G(G (n, m)) has the following
form.
···
G(G (n, m)) :
|
···
{z
}
2m edges
|
{z
}
n pyramidal vertices
The reason for the low linkedness of G (n, m) is that there are few vertices
that can be used on a “detour” for a linkage between the 2m pairs that are
not connected by an edge.
To determine the linkedness of G (n, m), we determine the linkedness of
graphs of type Kp \ M , where M is a matching. Obviously, the graphs of
G(G (n, m)) are of this type.
Lemma 5.4.5. Let G = Kp \ M , where M is a matching of size q ≤ p/2,
that is, G consists of q disjoint edges and p − 2q isolated vertices. Then the
linkedness of G is
( ¥p¦
,
if p ≤ 3q − 1
3
k(G) =
¦
¥ p−q
,
if p ≥ 3q − 1.
2
86
Proof. Observe that the connectivity of G is
½
p,
if q = 0
κ(G) =
p − 1,
if q ≥ 1,
and that G has a complete subgraph of size p − q. By Lemma 5.4.1, we
thus have
º ¹
º¾
½¹
p−q
κ(G)
,
.
(5.3)
k(G) ≥ min
2
2
Let p ≥ 3q − 1. Then choose all q pairs of nonconnected vertices and
⌋ pairs arbitrarily from the vertices of full degree. Then
additional ⌊ p−3q+2
2
these cannot be linked in G and the linkedness satisfies
º ¹
º
¹
p−q
p − 3q
=
.
k(G) ≤ q +
2
2
For q ≥ 1, we have ⌊(p − q)/2⌋ ≤ ⌊κ(G)/2⌋, and for q = 0, we have
⌊κ(G)/2⌋ = ⌊p/2⌋, and thus Equation 5.3 implies that also k(G) ≥ ⌊(p −
q)/2⌋.
Let p ≤ 3q − 1. Then choose ⌊p/3⌋ + 1 of the q nonedges of G as pairs.
These cannot be linked in G, as there are at most
p − 2(⌊p/3⌋ + 1) ≤ ⌊p/3⌋
additional vertices. Clearly, any ⌊p/3⌋ pairs of vertices of G can be linked,
as any such pair needs at most one additional vertex to be connected, and
if a pair is a nonedge, then every other vertex can be used.
For p = 3q − 1, both terms yield the same value.
Remark 5.4.6. For every p ∈ {3q − 2, 3q − 1, 3q, 3q + 1} we have that the
terms ⌊p/3⌋ and ⌊(p − q)/2⌋ yield the same value. Thus the case distinction
in Lemma 5.4.5 can be made at any of these boundary cases. We have
chosen the value p = 3p − 1 to be consistent with Gallivan [45] in the
statement of Corollary 5.4.10 below.
Lemma 5.4.7. Let n, m ≥ 0 be integers. The linkedness of G (n, m) is
given by
( ¥ 4m+n ¦
,
if n ≤ 2m − 1
3
k(G (n, m)) = ¥ 2m+n
¦
,
if n ≥ 2m − 1.
2
If we use substitutions (5.1) and (5.2), this evaluates to
( ¥ d+γ+1 ¦
,
if d + 2 ≤ 5γ
3
k(G (n, m)) = ¥ d−γ+1
¦
,
if d + 2 ≥ 5γ.
2
87
Proof. This follows from Lemma 5.4.5, as the number of vertices of G (n, m)
is 4m + n and the number of nonedges is 2m.
Example 5.4.8. Let d = 8 and γ = 2. Then n = 3 and m = 2 and we
obtain the 8-polytope on 11 vertices
P := G(3, 2) = ∆2 ∗ ¤ ∗ ¤ = pyr3 (¤ ∗ ¤).
The complement of the graph of P consists of 4 disjoint edges and 3 isolated
vertices. Obviously, G(P ) is not 4-linked. The polytope P is the smallest
known example of a polytope that is not ⌊d/2⌋-linked.
In combination with Lemma 5.4.3 we obtain the following result.
Theorem 5.4.9. Let d ≥ 2 and 0 ≤ γ ≤ (d + 2)/5. Then
¹
º
d−γ+1
k(d, γ) =
.
2
Choosing γ = ⌊(d+2)/5⌋, we obtain Gallivan’s examples, and the bound
of the last theorem implies the following bound on k(d) first given by Gallivan [45].
Corollary 5.4.10 (Gallivan [45]). The minimal linkedness of d-polytopes
satisfies
k(d) ≤ ⌊(2d + 3)/5⌋ .
One can construct polytopes with f0 = 3⌊d/2⌋ − 1 vertices that are not
⌊d/2⌋-linked. If d is even let
P := ∆2 ∗ ¤ ∗ ¤ ∗ C3∆ ∗ · · · ∗ C3∆ .
{z
}
|
m times
Then d = 4m + 8, f0 = 6m + 11 and k(P ) = 2m + 3. For d odd let
P := ∆4 ∗ ¤ ∗ ¤ ∗ ¤ ∗ C3∆ ∗ · · · ∗ C3∆ .
{z
}
|
m times
Then d = 4m + 13, f0 = 6m + 17 and k(P ) = 2m + 5.
For large f0 the methods used in this section to prove upper bounds fail.
This suggests the following question
Problem 5.4.11. Are all d-polytopes on at least 3⌊d/2⌋ vertices ⌊d/2⌋linked? Weaker: Is there some N (d), such that every d-polytope on at least
N (d) vertices is ⌊d/2⌋-linked?
88
Figure 5.4: Is this a subgraph of the complement graph of some 8-polytope on
12 vertices?
We know of only one obstruction for d-polytopes to not be ⌊d/2⌋-linked,
the obstruction exploited in this section: The polytopes G (n, m) have many
missing edges and not enough vertices to route all paths around the missing edges. If a polytope has 3⌊d/2⌋ or more vertices, there has to be a
different obstruction if it is not ⌊d/2⌋-linked. Regarding Problem 5.4.11,
it would be interesting to know if the graph in Figure 5.4 is a subgraph of
the complement graph of an 8-polytope on 12 vertices. The complement of
this graph is 8-connected, has at every vertex a subdivision of K9 rooted at
that vertex, and is not 4-linked.
It would also be interesting to determine the linkedness of cubical polytopes, that is, of polytopes whose facets all are (d − 1)-dimensional cubes.
Question 5.4.12. Is every cubical d-polytope ⌊d/2⌋-linked?
Start with the d-cube. It this ⌊d/2⌋-linked? If yes, then by Lemma 5.4.1,
every cubical d-polytope is at least ⌊(d − 1)/2⌋-linked. In particular, this
would imply a positive answer to Question 5.4.12 for odd dimensions d.
Also, a cubical polytope has at least 2d vertices by a result by Blind &
Blind [25], so a positive answer to Question 5.4.12 would also follow from
a positive answer to Problem 5.4.11 (if, in the weaker version, we have
N (d) ≤ 2d , of course).
5.4.3
Polytopes that Meet the Lower Bound
The main theorem of this section is Theorem 5.4.16. It states that polytopes
that meet the lower bound of Lemma 5.4.3 are
• unique, if f0 is odd, and
• of rather restricted combinatorial type, otherwise.
Furthermore, a lower bound on the number of combinatorial types in the
latter case is proven.
The “road map” to Theorem 5.4.16 is the following: We prove a concise
characterization of polytopes that have small facet complements in Theorem 5.4.14. We have not seen this characterization in work of others,
89
but suspect that it is known. This characterization implies the well-known
characterization of d-polytopes on d + 2 vertices [51, pp. 97–101].
This theorem already imposes severe conditions on the combinatorial
types of polytopes that meet the lower bound. The combinatorial type is
then even more restricted by Lemma 5.4.15, which analyzes the graphs of
the types that appear in Theorem 5.4.14.
The proof of Theorem 5.4.16 then mainly consists of the construction of
combinatorial types for the case when f0 is even.
We consider polytopes with small facet complements, that is, polytopes
in which every facet almost contains all vertices. If the size of every facet
complement is one, then clearly the polytope in question is the simplex.
The following lemma is the next step: What combinatorial types are
possible if for every facet there are at most 2 vertices that are not contained
in it?
Lemma 5.4.13. Let P be a d-polytope such that every facet F of P satisfies
| V(P ) \ V(F )| ≤ 2.
Then P is of the form
G (n, j1 , k1 , . . . , jm , km ) = ∆n−1 ∗ (∆j1 ⊕ ∆k1 ) ∗ · · · ∗ (∆jm ⊕ ∆km )
where k1 . . . , km , j1 , . . . , jm ≥ 1, and d = n − 1 + j1 + k1 + . . . + jm + km + m.
Proof. The property | V(P ) \ V(F )| ≤ 2 implies that the hypergraph of
facet-complements, that is, the hypergraph
Gcofacet (P ) := (V(P ), {W ⊆ V(P ) : V(P ) \ W is the vertex set of a facet})
is a graph (with no parallel edges, but possibly with loops). The edges of
Gcofacet (P ) are in bijection with the facets of P . Since the combinatorial type
of a polytope is determined by the vertex-facet incidences, the combinatorial
type of Gcofacet (P ) determines the combinatorial type of P .
For Q = G(n, j1 , k1 , . . . , jm , km ) the graph Gcofacet (Q) is a disjoint union
of n copies of the graph that consists of one single vertex and one single
loop, and complete bipartite graphs Kj1 ,k1 , . . . , Kjm ,km . Thus we have to
show that Gcofacet (P ) is of this type.
It is easy to see that loops can only occur at isolated vertices and that
there are no vertices of degree 1 in Gcofacet (P ) (we follow the convention
that loops contribute two edges to the degree count). Then it suffices to
check the following two properties of Gcofacet (P ):
(i) The graph Gcofacet (P ) does not have odd cycles (except loops at isolated vertices).
90
(ii) Whenever there is a path v1 v2 v3 v4 of length 3 in Gcofacet (P ), then v1 v4
is also an edge of Gcofacet (P ).
In fact, (i) follows from (ii) and the absence of triangles, as any larger odd
cycle together with (ii) would imply the existence of a triangle.
We now show that Gcofacet (P ) does not have triangles. Suppose there is
a triangle with vertices v1 , v2 , v3 and edges corresponding to facets F1 , F2 , F3
with V(F1 ) = V(P ) \ {v2 , v3 }, V(F2 ) = V(P ) \ {v1 , v3 }, and V(F3 ) = V(P ) \
{v1 , v2 }. Let F ′ be the face F1 ∩ F2 = F1 ∩ F3 = F2 ∩ F3 . Then clearly
F1 = pyrv1 (F ′ ), F2 = pyrv2 (F ′ ), and F3 = pyrv3 (F ′ ). Thus dim F ′ = d − 2
and P/F ′ is a 1-polytope on 3-vertices, a contradiction. (In face lattice
terms, this is a contradiction to the diamond property.)
Finally, we show that a path v1 v2 v3 v4 of length 3 implies the existence
of the edge v1 v4 . The edges of the path v1 v2 v3 v4 correspond to facets F1 , F2 ,
and F3 with V(F1 ) = V(P ) \ {v1 , v2 }, V(F2 ) = V(P ) \ {v2 , v3 }, and V(F3 ) =
V(P ) \ {v3 , v4 }. Let F ′ = F1 ∩ F2 ∩ F3 . Then clearly F ′ has dimension d − 3,
as only 4 vertices of P do not lie on F ′ .
The facets F1 , F2 and F3 are of dimension d−1 and each of them contains
exactly two more vertices than F ′ . We conclude that pyrv1 (F ′ ), pyrv2 (F ′ ),
pyrv3 (F ′ ), and pyrv4 (F ′ ) are all faces of P . Thus, P/F ′ is a 2-polytope on 4
vertices, which implies that F4 := pyrv4 (pyrv2 (F ′ )) is also a facet of P with
V(F4 ) = V(P ) \ {v1 , v4 }.
The proof is purely combinatorial, so one may ask to what extent the
statement generalizes. We have used the diamond property of the face
lattice. This is a stronger condition than relatively complementedness, and
indeed, the statement is false for graded relatively complemented lattices;
see for example the lattice in Figure 8.5(c).
We can define a subclass of the graded relatively complemented lattices
by requiring the diamond property and the property that every interval of
length 3 on 4 atoms looks like the lattice of a 4-gon. The proof generalizes
to this class (true?).
With Lemma 5.4.13, the following theorem is easily proven.
Theorem 5.4.14. Let d ≥ 2, let 0 ≤ γ ≤ d − 2, and let P be a d-polytope
on d + γ + 1 vertices. Then the following are equivalent:
(i) Every facet F of P satisfies | V(P ) \ V(F )| ≤ 2.
(ii) With m = γ, the polytope P is of the form
G (n, j1 , k1 , . . . , jm , km ) = ∆n−1 ∗ (∆j1 ⊕ ∆k1 ) ∗ · · · ∗ (∆jm ⊕ ∆km )
with suitable parameters ji and ki .
91
(iii) The polytope P does not have a simplex face of dimension d − γ + 1.
Proof. If | V(P ) \ V(F )| ≤ 2 for every facet F of P , then by Lemma 5.4.13
the polytope P is of the form G (n, j1 , k1 , . . . , jm , km ). Clearly, we have that
m = γ.
Now, suppose P is an iterated pyramid over a join of sums of simplices.
Let S be a simplex face of P of maximal dimension. Then S is the join of
∆n−1 with facets from each factor ∆ji ⊕ ∆ki . A facet of this sum in turn is
obtained by leaving out a vertex from each of the two simplices. Hence, S
has
n + j1 + k1 + . . . + jm + km = d − m + 1 = d − γ + 1
vertices and therefore dimension d − γ.
Finally, if P does not have a simplex face of dimension d − γ + 1, then
| V(P ) \ V(F )| ≤ 2 for every facet F . Otherwise, suppose there is a facet
F with | V(P ) \ V(F )| ≥ 3. Then γ(F ) ≤ γ − 2, and by Lemma 5.4.2 the
facet F has a simplex face of dimension
(d − 1) − γ(F ) = d − (γ(F ) + 1) ≥ d − γ + 1,
in contradiction to the hypothesis.
Theorem 5.4.14 contains the classification of d-polytopes on d + 2 vertices; see Grünbaum [51, pp. 97–101]: No d-polytope on d + 2 vertices
contains a simplex d-face. Thus, all polytopes on d + 2 vertices are of type
G (n, j1 , k1 , . . . , jm , km ) with m = γ = 1.
As for Lemma 5.4.2, there is also a Gale duality proof of Theorem 5.4.14
(this is left as an exercise for the reader).
So far, we have only excluded large simplex faces. The following lemma
analyzes the situation if we also exclude large subgraphs.
Lemma 5.4.15. Let P be a d-polytope on d + γ + 1 vertices. Suppose that
the graph G(P ) does not have a Kd−γ+2 -subgraph. Then P is of the form
G (n, m) = ∆n−1 ∗ |¤ ∗ ·{z
· · ∗ ¤},
m times
with n = d − 3γ + 1 and m = γ.
Proof. Since P does not have a Kd−γ+2 -subgraph, P does not have a simplex
face of dimension d − γ + 1. Thus, by Theorem 5.4.14, P is of the form
G(n, j1 , k1 , . . . , jm , km ).
92
To show that j1 = k1 = . . . = jm = km = 1 observe that the graph

 is the complete graph Kj+k+2 if j, k ≥ 2
if j ≥ 2, k = 1 or j = 1, k ≥ 2
G(∆j ⊕∆k ) contains a Kj+k+1

is a 4-cycle
if j = k = 1.
Furthermore, in a join P ∗ Q every vertex of P defines an edge with every
vertex of Q. Suppose now that ji ≥ 2 or ki ≥ 2 for some i. Then G(P )
contains a complete graph on
n + j1 + k1 + . . . + ji + ki + 1 + . . . + jm + km = d − m + 2 = d − γ + 2
vertices, which contradicts the hypothesis.
Theorem 5.4.16. Let d ≥ 2, and γ ≥ 0 with d ≥ 3γ − 1. Set n :=
d − 3γ + 1 ≥ 0, and m := γ. Let P be a d-polytope on f0 = d + γ + 1 vertices
with linkedness k(P ) = ⌊(d − γ + 1)/2⌋.
If f0 is odd, then P is of type
G (n, m) = ∆n−1 ∗ |¤ ∗ ·{z
· · ∗ ¤} .
m times
If f0 is even, then there are at least 2d − 3γ + 1 possibilities for the
combinatorial type of P . More precisely, exactly one of the following cases
applies:
(i) The polytope P is of type G (n, m).
(ii) We have m ≥ 1 and P is of type
G(n − k, 1, . . . , 1, k + 1),
| {z }
2m − 1 times
with 1 ≤ k ≤ n. In this case there are exactly n = d − 3γ + 1
possibilities for the type of P .
(iii) We have m ≥ 2 and P has a facet F of type
∆n+4 ∗ |¤ ∗ ·{z
· · ∗ ¤} .
m − 2 times
In particular, k(F ) = k(P ). In this case there are at least d − 1
possibilities for the combinatorial type of P .
93
Proof. If P is a d-polytope with d ≥ 2 on d + γ + 1 vertices and k(P ) =
⌊(d − γ + 1)/2⌋, then clearly d > γ ≥ 0, as every such polytope is at least
1-linked.
Let f0 be odd, that is, d − γ is even. If k(P ) = ⌊(d − γ + 1)/2⌋, then
G(P ) cannot have a Kd−γ+2 -subgraph, and by Lemma 5.4.15 the polytope
P is of type G (n, m) with m = γ and n = d + γ + 1 − 4γ = d − 3γ + 1.
Let f0 be even, that is, d − γ is odd. Under this assumption, the graph
of P cannot have a Kd−γ+3 -subgraph, as it would otherwise be higher linked
than ⌊(d − γ + 1)/2⌋. We distinguish the following three cases:
(1) The graph G(P ) does not have a Kd−γ+2 -subgraph.
(2) The graph G(P ) does have a Kd−γ+2 -subgraph, but P does not have a
(d − γ + 1)-dimensional simplex face.
(3) The polytope P has a (d − γ + 1)-simplex face.
These are all possible cases, and clearly they exclude each other.
Case (1). In this case, Lemma 5.4.15 implies that P is of type G (n, m).
Case (2). Under the assumptions in (2), Theorem 5.4.14 implies that the
polytope P is of type G(n′ , j1 , k1 , . . . , jm , km ) for m = γ ≥ 1 and suitable
parameters n′ , j1 , k1 , . . . , jm , km .
The complement graph G(P ) consists of some number q of disjoint edges
and f0 − 2q isolated vertices. Since G(P ) does not have a Kd−γ+3 subgraph
we must have f0 − q ≤ d − γ + 2, that is,
q ≥ 2γ − 1.
The only polytope of type G(n′ , j1 , k1 , . . . , jm , km ) with 2γ disjoint edges,
which is the maximum possible value for q, is G (n, m). However, G (n, m)
does not have a (d − γ + 1)-simplex face and thus we have q = 2γ − 1. This
is only achievable if, up to symmetry, j1 = k1 = · · · = jm−1 = km−1 = 1,
jm = 1 and 2 ≤ km ≤ d − 3γ + 2, that is, the first m − 1 factors of the join
are quadrilaterals and the last factor is a bipyramid. Thus P is of type
G(n − k, 1, . . . , 1 , k + 1),
| {z }
2m−1 times
for 1 ≤ k ≤ n = d − 3γ + 1. This gives d − 3γ + 1 different combinatorial
types for the different choices of k.
Case (3). In the final case, the polytope P has a (d − γ + 1)-simplex
face but not a (d − γ + 2)-simplex face.
Then for every facet F of P we have | V(P )\V(F )| ≤ 3. Otherwise there
is a facet F ′ , which is of dimension d − 1, that has γ(F ′ ) ≤ γ − 3. That is,
F ′ contains a simplex face of dimension (d − 1) − (γ − 3) = d − γ + 2, in
contradiction to the hypothesis.
94
By Theorem 5.4.14, there exists a facet F with | V(P ) \ V(F )| = 3, since
otherwise P would not have a (d − γ + 1)-simplex face. This facet must
satisfy | V(F ) \ V(F ′ )| ≤ 2 for every facet F ′ of F . Otherwise, we could
show with a similar calculation as before that there is a ridge F ′ of P with
a simplex face of dimension at least d − γ + 2. This implies that m = γ ≥ 2.
By Lemma 5.4.1, we have k(F ) ≤ k(P ). But also
º ¹
º
¹
d−γ+2
(d − 1) − (γ − 2) + 1
=
= k(P ),
k(F ) ≥
2
2
by Lemma 5.4.3 since d − γ + 2 is odd. Furthermore, (d − 1) − γ(F ) =
d − 1 − (γ − 2) = d − γ + 1 is even. Thus applying (i) yields that F is of
type G (n′ , m′ ) with n′ = d − 3γ + 6 and m′ = γ − 2, as
dim(G (n′ , m′ )) = n′ − 1 + 3m′ = d − 3γ + 5 + 3γ − 6 = d − 1.
To get the d − 1 different combinatorial types we modify G(n, m). Let
v0 , v1 be the vertices of one of the missing edges of one of the quadrilaterals.
Let Q := conv(V(G(n, m)) \ {v0 , v1 }). Let F be a k-face of Q that has a
quadrilateral 2-face, for 2 ≤ k ≤ d. Such a face exists since m = γ ≥ 2.
Take the subdirect sum R := (Q, F )⊕(∆1 , ∆1 ). Then the k-face F ⊕∆1 lies
in a facet of R whose complement has size 3, as F ⊕ ∆1 has a (k − 1)-face
that has a complement of size 3 in F ⊕ ∆1 .
For p = f0 and q = 2m, the polytope G (n, m) has linkedness
¹
º ¹
º
p−q
d−γ+1
k(G (n, m)) =
=
.
2
2
This implies that p ≥ 3q − 2 by Lemma 5.4.5; compare Remark 5.4.6.
Since d − γ + 1 is even, also p − q is even. The constructed polytope has
p vertices and exactly q − 1 disjoint missing edges. Lemma 5.4.5 implies
that the linkedness of G(R) is
¹
º ¹
º
p − (q − 1)
p−q
=
= k(G (n, m)),
2
2
since p − q + 1 is odd and p ≥ 3(q − 1) − 1.
As we have shown, for γ ≤ (d+2)/5 and f0 odd, polytopes that meet the
lower bound are combinatorially unique. Due to their special combinatorial
type, an iterated pyramid over a join of quadrilaterals, these polytopes are
projectively unique; see McMullen [77].
See Figure 5.5 for an illustration of minimal linkedness of 25-dimensional
polytopes.
95
k
∆25
13
simplicial polytopes
∆22 ∗ ¤
12
⌊d/2⌋
11
10
∆19 ∗ ¤ ∗ ¤
9
⌋
⌊ d+2
3
∆16 ∗ ¤ ∗ ¤ ∗ ¤
8
0
1
2
3
4
5
6
7
8
γ
9
Figure 5.5: Linkedness of polytopes in dimension 25. Shown in this graph are
(a) the minimal linkedness of simplicial polytopes, (b) the minimal linkedness of
polytopes on few vertices (in dimension 25 known up to γ = 5), (c) the unique
extremal examples for γ = 0, 1, 3, 5, and (d) the best known upper and lower
bounds for general polytopes.
In the case when f0 is even, the number of combinatorial types that are
tight for the bound increases at least linearly in d. It is a consequence of
Perles’ Skeleton Theorem [89] [62] that the number of graphs of polytopes
on d + γ + 1 vertices, however, is bounded by a function of γ; see Part II.
5.5
Minimal Linkedness in Small Dimensions
Theorem 5.3.2 and Corollary 5.4.10 imply the values for the minimal linkedness k(d) as displayed in Table 5.1.
In particular, we obtain exact values in dimensions 7, 10, and 13. The
value k(8) = 3 follows from Larman & Mani’s old lower bound [66] and
Gallivan’s upper bound [45].
d
k(d)
1
1
2
1
d
k(d)
11
4,5
12
4,5
3
1
4
2
13 14
5 5,6
5
2
6
2,3
15
5, 6, 7
...
...
7 8
3 3
9 10
3,4 4
Table 5.1: Possible values of k(d) in dimensions 1 ≤ d ≤ 15.
96
5.5. Minimal Linkedness in Small Dimensions
The value k(6) is the first open value of k(d) and it seems to be a difficult
problem to determine it. Our analysis of polytopes with few vertices in
Theorem 5.4.16 shows that k(6, 0) = k(6, 1) = k(6, 2) = 3. We have also
verified enumeratively that k(6, 3) = 3; beyond that we do not know much.
Problem 5.5.1. Determine k(6): Either show that all 6-polytopes are 3linked, or give an example of a 6-polytope P with k(P ) = 2.
Thomas & Wollan [113] have shown that every 6-connected graph with
at least 5f0 − 14 edges is 3-linked. For polytopes this implies that the
graph obtained from the graph of a 6-polytope by triangulating all 2-faces
arbitrarily is 3-linked, according to the Lower Bound Theorem for general
polytopes by Kalai [59]. Thus, all 2-simplicial 6-polytopes, that is, polytopes with only triangular 2-faces, are 3-linked.
To determine k(d) in general it suffices to look at iterated pyramids
over unneighborly polytopes, that is, polytopes in which every vertex lies on
an edge of the complement graph. This was remarked by McMullen [78].
Indeed, suppose that a polytope P has a vertex of full degree that is not
pyramidal. Then the pyramid over the link of that vertex yields the combinatorial type of a polytope that is at most as linked as P , as the resulting
graph is a subgraph of the graph of P on the same number of vertices.
However, the class of unneighborly polytopes contains the class of simple
polytopes (except for the simplex) and not even for them the exact values
of minimal linkedness are known. Indeed, the combinatorial structure of
unneighborly polytopes is not well-understood. For example, not even the
minimum number of vertices that an unneighborly polytope can have is
known exactly; see Part III.
97
Part II
Chapter 6
Skeleta of Polytopes with
Few Vertices
Perles’ Skeleton Theorem, a beautiful result by Micha A. Perles in the
theory of polytopes with few vertices, is discussed in this chapter and proved
in the following two. It is from an unpublished manuscript written by Perles
around 1970 [89], reported on and reproved by Kalai [62].
We will apply this theorem to problems by Marcus [71] and Bienia & Las
Vergnas [23, Exercise 9.35(iv)*]; see Problems 10.4.1 and 11.0.1 in Chapters 10 and 11, respectively.
Why are three entire chapters of this thesis devoted to this theorem?
There are a number of reasons:
(i) Perles’ original account on this theorem seems to be lost. I asked him
about it during a visit at the Hebrew University of Jerusalem in the
spring of 2007. He said that he did not have it, or rather, dass er es
nicht habe, as he speaks perfect German [88].
(ii) Kalai’s proof in [62] establishes an interesting connection to the ErdősRado Sunflower Lemma [40]. Unfortunately, the exposition is rather
sketchy and contains some inaccuraries. It also skips some key observations. (For example, it is not proved that a polytope on few vertices
does not have sunflowers of empty pyramids of “large” size. The proof
of this fact requires that the core of a sunflower is a face, which is not
quite as obvious as in the simplicial case.) Most importantly, the
bound on the number of vertices in missing pyramids given cannot be
101
6. Skeleta of Polytopes with Few Vertices
used as stated, since it depends on a parameter r that needs to be
guessed from the context. This, however, is the result we need for our
applications.
(iii) While none of the proofs given in Chapters 7 and 8 reaches the elegance of Kalai’s “sunflower proof,” our methods yield better bounds
on the number of missing pyramids than the bound implied by Kalai’s
argument.
(iv) The generalizations to strong PL spheres, graded relatively complemented lattices, and pyramidally perfect lattices we prove in Chapter 8 were so far not substantiated in print. As stated by Kalai
in [62, Section 2.4], his sunflower proof generalizes to the setting of
graded relatively complemented lattices, but this does not seem to be
straightforward and needs care in some touchy details. According to
Kalai [62, Section 2.4], Perles’ original proof also applies to this much
more general setting; but see (i). For one of our applications, namely
Problem 11.0.1, we need at least the level of generality of strong PL
spheres.
I believe these reasons warrant the time and effort spent on reproving
this theorem in the following two chapters. (Besides this, I had a lot of
fun doing it.) In this chapter, we give an introduction to, and overview of,
Perles’ Skeleton Theorem and its ramifications.
This is the essence:
Theorem 6.0.1 (Perles’ Skeleton Theorem [89] [62]). For parameters
k and γ, the number of combinatorial types of k-skeleta of d-polytopes on
d + γ + 1 vertices is bounded by a constant independent of the dimension d.
The key step in proving this result is to bound the number of empty
pyramids in the k-skeleton.
While bounds on the number of empty simplices in simplicial polytopes
were worked out exactly in a brilliant paper by Nagel [84], the general case
has so far not been given much attention. As mentioned, Kalai gave a
proof [62] that relies on the concept of sunflowers from the famous ErdősRado Sunflower Lemma [40].
We will discuss in Section 6.2 that, for polytopes with few vertices,
(a) bounding the number of combinatorial types of k-skeleta independently
of d,
(b) bounding the number of empty pyramids in the k-skeleton independently of d, and
(c) bounding the dimension in which k-skeleta can be realized “up to taking
pyramids” independently of d,
102
6.1. Reconstruction of Skeleta
are essentially equivalent ways of stating Perles’ original result. We will
therefore refer to any such statement as Perles’ Skeleton Theorem.
Chapters 7 and 8 are then entirely devoted to the problem of bounding
the number of empty pyramids in varying generality and by varying means.
6.1
Reconstruction of Skeleta
In this chapter we need the two notions of “empty simplex” and “empty
pyramid.” We generalize these concepts in Chapter 7.
Definition 6.1.1 (Vertex-induced, empty simplex, empty pyramid). Let C be a polytopal complex, and let U ⊆ V(C) be a subset of
the vertices of C.
The induced subcomplex of C on U , denoted C[U ], is the polytopal complex C[U ] induced by the vertex set U . That is, it consists precisely of all
the faces F of C with V(F ) ⊆ U . If for some subcomplex C ′ of C we have
C ′ = C[U ] for some U , we say that C ′ is vertex-induced .
For k ≥ 1, an empty k-simplex of C is a vertex-induced subcomplex of
C that is isomorphic to the boundary of the k-simplex. Similarly, an empty
k-pyramid of C is a vertex-induced subcomplex of C that is isomorphic to
the boundary of a pyramidal k-polytope.
In the special case k = 1 we also speak of missing edges.
The combinatorial type of a simplicial polytope (or more generally, a
simplicial complex) is uniquely determined by its empty simplices: A set
of vertices forms a face if and only if it does not contain the vertices of an
empty simplex.
For general polytopal complexes, for example, boundary complexes of
nonsimplicial polytopes, knowing the empty simplices is not enough, as the
following example shows.
Example 6.1.2. Consider the 2-skeleta of the following two polytopes:
P1 := ∆7 ∗ ¤, and
P2 := bipyr(∆3 ) ∗ bipyr(∆3 ).
Both have f0 = 12 and the only empty simplices are two disjoint missing
edges: in P1 , these are the diagonals of the quadrilateral two face; in P2 ,
these are formed by the apex vertices of the bipyramids. In terms of empty
simplices the 2-skeleta are indistinguishable. But P1 has a quadrilateral
2-face, and P2 does not. The 2-skeleta are therefore not isomorphic.
103
In the former example, we could have remedied the situation by reconstructing the 2-skeleta from the complexes induced by the vertices in empty
simplices. The following example shows that this is not sufficient in general.
Example 6.1.3. Let Q := pyr(¤), the pyramid over a quadrilateral. Consider the 3-skeleta of
P1 := bipyr(Q) ∗ bipyr(∆4 ), and
P2 := Q ∗ bipyr2 (∆4 )
They both have the same number of vertices. In P1 , the boundary of Q
forms an empty 3-pyramid. In P2 , Q is a 3-face and there are no empty
3-pyramids. Thus, the 3-skeleta of P1 and P2 are not isomorphic.
Let v be the apex of Q. This does not lie in an empty simplex, neither
of P1 nor of P2 . However, the 3-skeleta of P1 and P2 induced on V(P1 ) \ {v}
and V(P2 ) \ {v}, respectively, are isomorphic. Consequently, we cannot
distinguish the 3-skeleta from the complexes induced by vertices in empty
simplices.
This example shows that we also need to consider vertices in empty
pyramids. Indeed, this is enough, as we establish in the following.
Definition 6.1.4 (Kernel). Let k ≥ 1 and let C be a polytopal complex of
dimension at least k. Let U be the set of vertices u of C that are contained
in some empty ℓ-pyramid with ℓ ≤ k.
We define the k-kernel Kerk (C) of C to be the k-skeleton of C[U ]. If a
polytopal complex C is a k-kernel of a polytope P with γ(P ) = γ we call C
a (k, γ)-kernel.
The following lemma is an important step towards Perles’ Skeleton Theorem. Although its proof is straightforward, the lemma illustrates the importance of the concept of empty pyramids.
Lemma 6.1.5 (Kalai [62]). The k-skeleton of a polytopal complex C can
be reconstructed, up to isomorphism, from Kerk (C) and the number f0 (C).
Proof. Let V be the set of vertices of C and A = V \ V(Kerk (C)). Every
vertex of A forms the apex of a pyramid over every face of dimension at
most k − 1. Thus the set of faces of skelk (C) is precisely
{F ∗ B : F ∈ Kerk (C), B ⊆ A, dim(F ) + |B| ≤ k}.
To reconstruct the complex up to isomorphism it clearly suffices to know
Kerk (C) and the number of vertices.
104
6.2. Perles’ Skeleton Theorem
This lemma can also be stated as follows: The k-skeleton of C “looks”
like the k-skeleton of the join of Kerk (C) with a simplex of dimension
f0 (C) − | V(Kerk (C))| − 1.
Thus, one important idea in connection with Perles’ Skeleton Theorem
is to distinguish skeleta of polytopes only “up to taking pyramids in the
k-skeleton.”
Definition 6.1.6 (Pyramidally equivalent). Let C 1 and C 2 be polytopal
complexes in Rd . We say that C 1 and C 2 are pyramidally k-equivalent if they
have isomorphic k-kernels.
If two complexes C 1 and C 2 are pyramidally k-equivalent, then there are
nonnegative integers n1 and n2 such that
skelk (pyrn1 (C 1 )) ≃ skelk (pyrn2 (C 2 )).
Without loss of generality we can even assume that n1 = 0 or n2 = 0.
We give complete lists of pyramidally inequivalent skeleta of polytopes
for small parameters in Section 7.4.
6.2
In this section we state Perles’ Skeleton Theorem in different guises. We
write down three statements, Perles’ Skeleton Theorem I, II, and III. We
justify calling all of them Perles’ Skeleton Theorem by showing that they
are essentially equivalent: Each one of them can easily be deduced from
any one of the others. The ideas are from Kalai’s paper [62].
6.2.1
The Number of Empty Pyramids
We start with the number of empty pyramids. This is the statement we
prove in Chapter 7 for polytopes and in Chapter 8 for more general objects.
Theorem 6.2.1 (Perles’ Skeleton Theorem I). For any d-polytope P
on d + γ + 1 vertices, the total number of empty ℓ-pyramids with ℓ ≤ k is
bounded by a function of k and γ.
If k ≥ 1 and C is a polytopal complex, we denote by M (k, C) the total
number of empty ℓ-pyramids of C with ℓ ≤ k. Furthermore, we define
M (k, γ) := max{M (k, P ) : P ∈ P γd }
d≥0
that is, M (k, γ) is the maximum number of empty ℓ-pyramids with ℓ ≤ k
of any polytope P with γ(P ) = γ. This is well-defined by Theorem 6.2.1.
105
6.2.2
The Number of Combinatorial Types of Skeleta
Lemma 6.2.2. Let P be a d-polytope on d+γ +1 vertices and k ≥ 1. Every
empty k-pyramid of P has at most k + γ vertices.
Proof. Clearly, a (k − 1)-face G of P can have at most k + γ vertices, with
equality if and only if P is a (d − k + 1)-fold pyramid over G. The statement
follows.
For fixed γ, let N be the maximum number of vertices in the k-kernel
of a polytope with γ(P ) = γ. Since the number of empty pyramids of C
is bounded by a function of k and γ, also the size of N is bounded by a
function of k and γ. Indeed, by Lemma 6.2.2 we have N ≤ (k + γ)M (k, γ).
The number of k-kernels of d-polytopes on d + γ + 1 vertices is then
bounded by the number of k-dimensional polytopal complexes on N vertices
such that each k-face has at most k+γ vertices. And this number is bounded
by
exp2
k+γ µ ¶´
³X
N
j=1
j
≤ exp2
¶
k+γ µ
³X
(k + γ)M (k, γ) ´
j=1
j
≤ exp2 (exp2 ((k + γ)M (k, γ))),
which is a function of k and γ.
By Lemma 6.1.5, the k-skeleton can be reconstructed from the k-kernel.
Thus, Theorem 6.2.1 implies the following theorem.
Theorem 6.2.3 (Perles’ Skeleton Theorem II). The number of combinatorial types of k-skeleta of d-polytopes on d + γ + 1 vertices is bounded
by a function of k and γ.
For integers γ ≥ 0 and k ≥ 1 let C(d, k, γ) be the number of combinatorial types of k-skeleta of d-polytopes on d + γ + 1 vertices. Because of
Theorem 6.2.3, it makes sense to define
C(k, γ) := max{C(d, k, γ)}.
d≥0
The function C(k, γ) counts the number of combinatorial types of k-skeleta
of polytopes P with γ(P ) = γ.
6.2.3
The Realizability Dimension of Kernels
Let D(k, γ) be the smallest dimension such that C(k, γ) = C(D(k, γ), k, γ).
For every e ≥ D(k, γ) we thus have C(e, k, γ) = C(k, γ).
106
6.2. Perles’ Skeleton Theorem
By taking pyramids, we can realize any k-kernel of a polytope in P γd as
the k-kernel of a polytope in Peγ . Thus the k-skeleta of polytopes in P γe are
pyramidally k-equivalent to k-skeleta of polytopes in Pdγ . We therefore call
D(k, γ) the realizability dimension of (k, γ)-kernels.
These observations show that Theorem 6.2.3 implies the following statement.
Theorem 6.2.4 (Perles’ Skeleton Theorem III). For integers k ≥ 1
and γ ≥ 0 there is d ≥ 0 such that for every e ≥ d the k-skeleton of an
e-polytope on e + γ + 1 vertices is pyramidally k-equivalent to the k-skeleton
of some d-polytope on d + γ + 1 vertices.
To bring this section to a close, we show that Theorem 6.2.4 implies
Theorem 6.2.1: The number of vertices in the k-kernel of any polytope
with γ(P ) = γ is bounded by D(k, γ) + γ + 1, which is a function of k and
γ. Thus, also the number of empty pyramids is bounded by a function of k
and γ.
Remark 6.2.5. We have also shown the following quantitative relations
between the functions M (k, γ), C(k, γ), and D(k, γ):
C(k, γ) ≤ exp2
¶
k+γ µ
³X
(k + γ)M (k, γ) ´
j
j=1
, and
¶
k+γ µ
X
D(k, γ) + γ + 1
.
M (k, γ) ≤
j
j=1
In particular, if we bound M (k, γ) we also have a bound for C(k, γ).
There seems to be no hope of finding an explicit bound for the function
D(k, γ). To do so, one would have to find a way to realize a given k-kernel
of a d-polytope on d + γ + 1 vertices in some dimension d′ (k, γ) ≥ D(k, γ).
Indeed, a simplicial polytopal complex can always be realized in a polytope
whose dimension is bounded by a function of the size of the complex; see [63,
Proposition 20.2.3]. However, with the known techniques one does not have
control over the size of γ.
Remark 6.2.6. By definition, the functions M (k, γ), C(k, γ), and D(k, γ)
are weakly monotone in k.
We list a few more properties:
• The function M (k, γ) is strictly monotone in γ. Take a polytope
that has M (k, γ) empty pyramids of dimension at most k and stack
107
a facet of this polytope. Then γ goes up by one, and M (k, γ + 1)
goes up at least by one. It is not, however, strictly monotone in k.
Every d-polytope on d + 2 vertices has at most 2 emtpy pyramids of
dimension at most k, and this bound is attainable for every k ≥ 1;
see Section 7.4.
• The function C(k, γ) is strictly monotone both in k and in γ. By
doing a vertex split at the vertex of an empty k-pyramid, we obtain
that C(k + 1, γ) > C(k, γ). To see that C(k, γ + 1) > C(k, γ), we
take a polytope P with γ(P ) = γ with a kernel of maximum size and
stack a facet.
• Monotonicity properties of the function D(k, γ) do not seem to be
easy to prove. We do however have a lower bound on the size of
D(k, γ). Lemma 5.4.2 and Theorem 5.4.14 imply that a polytope
P with γ(P ) = γ with the maximum number of 2γ disjoint empty
k-pyramids is necessarily of the form
P = (∆k ⊕ ∆k ) ∗ · · · ∗ (∆k ⊕ ∆k ) .
{z
}
|
γ factors in the join
Thus D(k, γ) ≥ dim(P ) = γ(2k + 1) − 1.
108
Chapter 7
Perles’ Skeleton Theorem for Polytopes
In this chapter we prove Perles’ Skeleton Theorem for polytopes by deriving
a bound on the number of empty pyramids of bounded dimension.
An empty pyramid has the important geometric property of being necessarily flat, in terminology by Richter-Gebert [94, p. 31]: It is flat, that
is, if we take the convex hull, we obtain a polytope of the same dimension,
and it is flat in every geometric realization—it is necessarily flat.
A general empty face, which is defined similarly to empty pyramids,
except that the combinatorial type may be arbitrary, may not be flat. For
example, induced cycles in the graph of a polytope are empty 2-faces, and
if such a cycle has length at least 4, it obviously might have an affine
dimension that is larger than 2; see Figure 7.1.
Why is the property of being flat important?
It is important for our inductive approach to proving Perles’ Skeleton
Theorem. We want to count the total number of empty pyramids, and we
do this by taking vertex figures and counting inductively.
A general empty face might “disappear” when we pass to a vertex figure; for an example consider the vertex figure at v of the empty 2-face in
Figure 7.1. This is not true for flat empty faces: A flat empty face yields
a flat empty face of one dimension less in a vertex figure, except for trivial
cases. We prove this in Section 7.1.
Indeed, counting flat empty faces seems to be the “right thing to do.”
Although knowing the empty pyramids is sufficient for reconstructing the
k-skeleton, they complicate the induction: An empty pyramid is not neces-
109
7. Perles’ Skeleton Theorem for Polytopes
v
Figure 7.1: The empty 2-face indicated by the bold lines is “invisible” in the
vertex figure at v.
sarily an empty pyramid in a vertex figure. To see this, consider a vertex
figure at the apex of an empty pyramid. The empty face in this vertex
figure is isomorphic to the boundary of the base of the empty pyramid. But
this might be of arbitrary combinatorial type.
One can get around this problem and argue that we can still count all
empty pyramids by considering them in the right quotients, but the result
is slightly clumsy; see Section 8.4.
It is more natural to simply count flat empty faces, and this is what we
do. Here is a rough outline of the proof:
• In Section 7.1, we establish basic properties of flat empty faces: Flat
empty faces yield flat empty faces in quotients, and two distinct flat
empty faces yield distinct flat empty faces in quotients. These two
properties make sure that we do not miscount the total number of flat
empty faces.
• In Section 7.2, we give the main parts of the proof: We bound the
number of missing edges, and then by induction the number of flat
empty k-faces. To accomplish this, we have to select a small number
of vertices such that we count all flat empty faces by counting flat
empty faces in vertex figures at these vertices. We apply Grünbaums’
theorem (Theorem 2.1.2) on simplex refinements to accomplish this.
The rest of the chapter consists of Section 7.4, where we look at pyramidally inequivalent complexes of polytopes for small parameters, and a
survey of results related to the simplicial version of Perles’ theorem; see
110
7.1. Empty Faces in Polytopes
Section 7.5.
7.1
Empty Faces in Polytopes
We generalize the concept of empty simplices and empty pyramids here.
Definition 7.1.1 (Empty face, empty pyramid, empty simplex). Let
C ⊂ Rd be a polytopal complex and k ≥ 1.
An empty k-face M of C is a vertex-induced subcomplex of C that is
combinatorially isomorphic to the boundary complex of a k-polytope P . If
P is a pyramid or a simplex, the empty face is called empty k-pyramid or
empty k-simplex,respectively, as defined before.
A flat empty k-face M is an empty k-face of C that equals as a polytopal
complex the boundary of the polytope conv(V(M )).
If M is an empty k-face, flat empty k-face, empty k-pyramid, or empty
k-simplex of C for some k, we say that M is an empty face, flat empty face,
empty pyramid, or empty simplex, respectively.
Definition 7.1.2 (Quotient of an empty face). Let P ⊂ Rd be a dpolytope, M an empty face of P , and let F be a face of M . Let G1 , . . . , Gm
be the faces of M that contain F , and consider a realization of the quotient
polytope P/F . The polytopal subcomplex of P/F given by the set of faces
{G1 /F, G2 /F, . . . , Gm /F } is called the quotient of the empty face M at F
and will be denoted by M/F .
7.1.1
Flat Empty Faces in Quotients
Lemma 7.1.3. Let d ≥ k ≥ 1, let P be a d-polytope, and let M be a flat
empty k-face of P . Then M is not contained in a k-face of P .
Proof. This is “geometrically clear”: We can assume that k = d, otherwise
we intersect P with the k-flat aff(M ). But P cannot have an empty dface.
Lemma 7.1.4. Let P be a d-polytope and let 2 ≤ k1 , k2 ≤ d − 1. For
i = 1, 2, let Mi be a flat empty ki -face. Suppose that M1 and M2 are
distinct but intersect in a vertex v. Then there is a realization of P/v such
that the following hold:
(i) For i = 1, 2, Mi /v is a flat empty (ki − 1)-face of P/v.
(ii) The flat empty faces M1 /v and M2 /v are distinct.
111
Proof. Let H be a hyperplane such that v is strictly on one side of H and
V(P ) \ {v} is strictly on the other side of H. Then the intersection P ∩ H
is a realization of P/v in H ∼
= Rd−1 . In the following we will denote this
realization simply by P/v.
Let i ∈ {1, 2}. All the proper faces of conv(Mi ∩H) are faces of P/v and,
since ki ≥ 2, there is at least one such face of dimension at least zero. We
first show that conv(Mi ∩ H) is not a face of P/v. Let Ki := aff(Mi ) ⊂ Rd
be the k-dimensional affine span of Mi . Because v lies in Mi , the affine
flat Ki intersects H in an affine (k − 1)-flat Ki′ := H ∩ Ki . Assume that
conv(Mi ∩ H) = Ki′ ∩ P/v is a (k − 1)-face of P/v. The polytope P is
contained in the cone with apex v that is spanned by P/v. The k-flat
Ki contains v and the (k − 1)-face conv(Mi ∩ H) of P/v. Therefore, Ki
intersects P in a k-face F . However, Mi is contained in Ki and therefore in
F , a contradiction to Lemma 7.1.3.
Clearly, if k1 6= k2 , then M1 /v and M2 /v are distinct. So suppose k1 =
k2 . If M1 /v and M2 /v are equal, then also K1 = K2 . Consequently, both
M1 and M2 lie in the k-flat K := K1 = K2 . By convexity M1 ⊂ M2 and, by
symmetry, M2 ⊂ M1 , so M1 = M2 , in contradiction to the hypothesis.
Corollary 7.1.5. Let P be a d-polytope and let 1 ≤ k1 , k2 ≤ d − 1. For
i = 1, 2, let Mi be a flat empty ki -face and suppose that M1 and M2 are
distinct. Let F be a face of M1 and M2 of dimension at most min{k1 , k2 }−2.
Then there is a realization of P/F such that the following hold:
(i) For i = 1, 2, Mi /F is a flat empty (ki − dim(F ) − 1)-face of P/F .
(ii) The flat empty faces M1 /F and M2 /F are distinct.
Proof. If min{k1 , k2 } = 1, the statement is trivial, as then F = ∅ and
P/F = P . Otherwise, apply Lemma 7.1.4 inductively.
7.2
A Bound on the Number of Flat Empty Faces
The proof of the following lemma is similar to a proof by Perles of a bound
on the number of missing edges [88].
Theorem 7.2.1. Let P be a d-polytope on d + γ + 1 vertices. Then the
number of missing edges is bounded by γ(γ + 1).
Proof. The statement holds for 2-dimensional polytopes, as the number of
missing edges of a 2-polytope on 3 + γ vertices is
µ
¶
γ(γ + 3)
3+γ
− (3 + γ) =
≤ γ(γ + 1),
2
2
112
7.2. A Bound on the Number of Flat Empty Faces
with equality only for γ = 0 and γ = 1.
Let P be a d-polytope on d + γ + 1 vertices with d ≥ 3. Choose a vertex
v ∈ V(P ), and let P/v be a vertex figure of P at v. Write γ ′ := γ(P/v). Let
the set D consist of the neighbors of v and v itself, and let B = V(P ) \ D.
Then |B| = γ − γ ′ .
By induction on the dimension, the number of missing edges in P/v is
at most γ ′ (γ ′ + 1).
Suppose that e′ is an edge of P/v whose endpoints, under radial projection from v, map to a missing edge of P . Then e′ corresponds to a 2-face of
P with at least 4 vertices. Thus, there is at least one vertex of this 2-face
that belongs to B. No two such 2-faces share vertices in B, and so the
number of such edges is at most |B| = γ − γ ′ .
Every other missing edge of P is incident to at least one vertex of B.
Since every vertex in a d-polytopes has degree at least d, the total number
of missing edges is then at most
γ ′ (γ ′ + 1) +
| {z }
missing edges
in P/v
as claimed.
γ − γ′
| {z }
edges missing
because of 2-faces
+ γ(γ − γ ′ ) ≤ γ(γ + 1),
| {z }
missing edges
incident to B
Remark 7.2.2. The bound of γ(γ + 1) is best possible. It is attained by
P = I × ∆γ , the prism over a γ-simplex. This is a simple polytope and
any vertex in {0} × ∆γ has exactly one neighbor in {1} × ∆γ . Thus any
vertex of the bottom face of P lies in exactly γ missing edges. The total
number of missing edges then is γ(γ + 1). Take pyrd−γ−1 (P ) to obtain a
d-polytope for which the bound on the number of missing edges is tight for
any d ≥ γ + 1.
Lemma 7.2.3. Let P be a d-polytope on d+γ+1 vertices and let φ : ∆d → P
be a refinement homeomorphism with affinely independent principal vertices
D ⊆ V(P ).
Then there are at most γ flat empty faces induced by vertices in D and
they are all empty simplices.
Proof. If M is a flat empty ℓ-face induced by vertices in D, that is, V(M ) ⊆
D, then the vertices of M are affinely independent. Clearly, M is an empty
ℓ-simplex on ℓ + 1 vertices.
If τ ⊆ ∆d is the ℓ-simplex with V(τ ) = φ−1 (V(M )), then φ(τ ) contains
a vertex besides the vertices of M . This must be a vertex in B := V \ D.
Since φ is a bijection, there can be at most |B| = γ flat empty faces, which
are necessarily all empty simplices.
113
The example of the refinement homeomorphism from the 3-simplex to
the regular 3-crosspolytope in which the four equatorial vertices are principal, compare Section 2.1, shows that the previous lemma fails for more
general refinement homeomorphisms.
Theorem 7.2.4. Let P be a d-polytope on d + γ + 1 vertices and 1 ≤ k ≤
d − 1. Then the number of flat empty ℓ-faces with ℓ ≤ k is bounded by
γ(γ + 1)k .
Proof. We prove the statement by induction on k. Let v ∈ V(P ) be a
vertex of P . Let φ : ∆d → P be a refinement homeomorphism rooted at v
with affinely independent principal vertices D = φ(V(∆d )). Furthermore,
let B = V(P ) \ D. The size of B is |B| = γ.
The case k = 1 is Theorem 7.2.1, so we have that the number of flat
empty 1-faces is bounded by γ(γ + 1).
Let k ≥ 2. The number of flat empty faces induced by vertices in D is
bounded by γ, according to Lemma 7.2.3.
Every other flat empty face contains at least one vertex in B. For w ∈ B
consider the vertex figure P/w and let γ ′ = γ(P/w). By induction, the
number of flat empty ℓ-faces with ℓ ≤ k − 1 of P/w is at most γ ′ (γ ′ + 1)k−1 .
By Lemma 7.1.4, this number bounds the number of flat empty ℓ-faces of
P with 2 ≤ ℓ ≤ k that are incident to w, as any two such flat empty faces
yield distinct flat empty faces in P/w.
The number of missing edges incident to w ∈ B is bounded by γ − γ ′ .
Therefore, we count at most γ ′ (γ ′ + 1)k−1 + γ − γ ′ ≤ γ(γ + 1)k−1 flat empty
ℓ-faces with ℓ ≤ k at w. This number bounds all flat empty ℓ-faces of P
with ℓ ≤ k that contain w.
The sum over all vertices in B, together with the number of flat empty
faces induced by vertices in D, yields a total bound of
γ + γ 2 (γ + 1)k−1 = γ(γ + 1)k ,
as claimed.
As every empty pyramid is a flat empty face we have also shown the
following.
Corollary 7.2.5 (Perles’ Skeleton Theorem I). Let k ≥ 1 and γ ≥
0. Then the total number of empty ℓ-pyramids with ℓ ≤ k is bounded by
γ(γ + 1)k .
114
7.3. Disjoint Empty Faces
7.3
Disjoint Empty Faces
In contrast to Theorem 7.2.4, we derive here a bound on the number of disjoint empty faces in a polytope. This is much simpler, we get a tight bound,
and we even have a characterization of polytopes that attain equality.
Lemma 7.3.1. Let P be a d-polytope on d + γ + 1 vertices and d ≥ 2. Then
P has at most 2γ vertex-disjoint empty faces.
Proof. By Lemma 5.4.2, the polytope P has a max{1, (d − γ)}-dimensional
simplex face S. Any empty face of P has a vertex outside S. Therefore,
there can be at most d + γ + 2 − (d − γ + 2) = 2γ disjoint empty faces.
The bound of 2γ is tight. It is attained by polytopes of the form
∆n ∗ (∆j1 ⊕ ∆k1 ) ∗ (∆j2 ⊕ ∆k2 ) ∗ . . . ∗ (∆jm ⊕ ∆km ),
where k1 . . . , km , j1 , . . . , jm ≥ 1, d = n + j1 + k1 + . . . + jm + km + m and
f0 = d + γ + 1 = n + 1 + j1 + k1 + . . . + jm + km + 2m, that is, γ = m.
There are exactly 2 disjoint empty faces (in this case empty simplices) in
each factor of the join. Thus altogether there are 2m = 2γ disjoint empty
faces. In Theorem 5.4.14 we have shown that these are the only polytopes
that attain the bound of 2γ.
Lemma 7.3.1 is an important ingredient of Kalai’s proof [62] of Perles’ Skeleton Theorem, which uses a variant of the Erdős-Rado Sunflower
Lemma [40].
Let X = {A1 , . . . , Am } be a collection of sets. The collection X is called
a sunflower of size m if G := ∩m
k=1 Ak = Ai ∩ Aj and Ai − G 6= ∅ for all
i, j ∈ {1, . . . , m}, i 6= j. The set G is called the core of the sunflower and
the sets Ai − G the petals of the sunflower.
Using Lemmas 7.3.1 and 8.2.3, and Corollary 8.2.10, one can show that
a d-polytope on d + γ + 1 vertices does not have a “sunflower of empty
pyramids” of size 2γ + 1 (consider the vertex sets of empty pyramids as a
set system).
A slight variation of the proof of the Erdős-Rado Sunflower Lemma [40]
yields that every collection of nonempty sets, each of which does not exceed
a given size, that does not have a “large” sunflower is “small”; compare
Kalai [62]. This gives another proof of Perles’ Skeleton Theorem. From
this argument one can derive a bound of (k + γ)!(2γ)k+γ on the number of
vertices in the k-kernel. (This proof does not generalize in an obvious way
to flat empty faces.)
115
7.4
Pyramidally Inequivalent Complexes
We give complete lists of (k, 1)-kernels and of (1, 2)-kernels to illustrate
Perles’ theorem.
7.4.1
Kernels of Polytopes on d + 2 Vertices
The classification of d-polytopes on d + 2 vertices [51, pp. 97–101], see also
Theorem 5.4.14, tells us that every such polytope P is given by parameters
k1 , k2 , d ∈ N with k1 + k2 ≤ d as
P = pyrd−(k1 +k2 +2) (∆k1 ⊕ ∆k2 ).
This polytope has exactly two missing faces: one missing k1 -simplex and
one missing k2 -simplex. Thus for every k only the following three types of
kernels appear:
(i) There is no empty face of dimension at most k, so the kernel is empty.
In this case, k1 , k2 ≥ k + 1.
(ii) There is exactly one empty face of dimension at most k. This corresponds to the case where, up to symmetry, k1 ≤ k and k2 ≥ k + 1.
(iii) There are two disjoint empty faces of dimensions k1 ≤ k and k2 ≤ k.
7.4.2
Kernels of Polytopes on d + 3 Vertices
Polytopes on d + 3 vertices correspond to 2-dimensional Gale diagrams. We
give a full list of kernels that appear for k = 1.
We need the following basic building blocks of graphs and the operation
of disjoint union to combine them:
Definition 7.4.1 (Elementary graphs, disjoint union). Let k ≥ 1. We
denote by E k the graph on 2k vertices with k edges (all of them disjoint),
by P k the path on k edges, and by C k the simple cycle on k edges.
We write G1 ∪· G2 for the disjoint union of graphs G1 and G2 under the
assumption that V(G1 ) ∩ V(G2 ) = ∅.
We have the following conditions on the number of missing edges:
• There are at most 4 disjoint missing edges, by Lemma 7.3.1. If there
are exactly 4 missing edges, then the combinatorial type of the polytope is determined, by Theorem 5.4.14.
• There are at most 6 missing edges in total, by Theorem 7.2.4.
Furthermore, we have the following structural conditions imposed by
the geometry of the Gale diagram:
116
7.4. Pyramidally Inequivalent Complexes
2
2
3
(a)
(c)
(b)
2
2
2
(e)
(d)
Figure 7.2: Gale diagrams of 4-polytopes on 7 vertices that realize 1-kernels of
polytopes on d + 3 vertices.
• The complement graph of a d-polytope on d + 3 vertices does not
contain a 3-cycle or a 4-cycle.
• If the complement graph contains a 5- or 6-cycle, then there are no
further missing edges.
• The graph E 1 ∪· P 4 is not the complement graph of a d-polytope on
d + 3 vertices.
(The proof of these conditions is left to the reader.)
The list in the following theorem is a complete list of graphs that do not
violate any of the above conditions. Thus, to proof the theorem we only
need to find a realization for each of the given kernels.
Theorem 7.4.2. The following is a complete list of (1, 2)-kernels of polytopes:
• E 1, E 2, E 3, E 4,
• P 2, P 3, P 4,
• E 1 ∪· P 2 , E 2 ∪· P 2 , E 1 ∪· P 3 .
117
• C 5, C 6.
Furthermore, the realizability dimension D(1, 2) of (1, 2)-kernels is 5, that
is, every (1, 2)-kernel is the 1-kernel of some 5-polytope on 8 vertices.
Proof. We go through the list and give a realization for each of the graphs
as a (1, 2)-kernel:
• Graph E 1 : This graph is not realizable as the 1-kernel of a 3-polytope
on 6 vertices, as K5 is not planar. However, it is realized by a 4polytope on 7 vertices, given by the Gale diagram in Figure 7.2(a).
• Graph E 2 : A graph on 6 vertices with E 2 as its complement contains
a K3,3 -subgraph, so this graph is not realizable as the 1-kernel of a
3-polytope on 6 vertices. It is realized by a 4-polytope on 7 vertices
with the Gale diagram in Figure 7.2(b).
• Graph E 3 : A 3-polytope on 6 vertices that realizes this graph as its
1-kernel is the octahedron.
• Graph E 4 : By Lemma 7.3.1, every d-polytope on d + 3 vertices with
this graph as its complement is an iterated pyramid over the join of
two quadrilaterals. The one with the smallest dimension clearly is the
join of two squares, that is, a 5-polytope on 8 vertices.
• Graph P 2 : The graph K6 \ P 2 has a K3,3 -subgraph, so P 2 is not the
1-kernel of a 3-polytope on 6-vertices. It is realized by a 4-polytope
given by the Gale diagram in Figure 7.2(c).
• Graph E 1 ∪· P 2 : Because of a K3,3 -subgraph, this is not the 1-kernel of
a 3-polytope on 6 vertices. It is realized by a 4-polytope on 7 vertices
given by the Gale diagram in Figure 7.2(d).
• Graph E 2 ∪· P 2 : This is realized as the 1-kernel of a 4-polytope on 7
vertices, given by the Gale diagram in Figure 7.2(e).
• This is realized by a 3-polytope on 6 vertices, a vertex split of a 5-gon.
• Graph E 1 ∪· P 3 : This is realized by a 3-polytope P on 6 vertices. Let Q
be a quadrilateral in R2 and a one of its edges. Let P := (Q, a)⊕(e, e)
for a line segment e.
• Graph P 4 : Let Q be a quadrilateral and take R := pyr(Q). Construct
a polytope by stacking onto one of the triangular faces of R. The
resulting 3-polytope on 6 vertices realizes P4 as a 1-kernel.
• Graph C 5 : This is realized by a 5-gon.
• Graph C 6 : This is realized by a prism over a triangle.
The largest dimension we needed was d = 5, and thus D(1, 2) = 5.
118
7.5. Empty Simplices in Simplicial Polytopes
7.5
Empty Simplices in Simplicial Polytopes
There is a wealth of results related to the simplicial version of Perles’ Skeleton Theorem by Kalai [62] and Nagel [84]. A short survey of these results
is given here.
Let P be a simplicial d-polytope. An empty pyramid of P is an empty
simplex, so M (k, P ) counts the number of empty ℓ-simplices of P with
ℓ ≤ k. Let
MS (k, γ) := max{M (k, P ) : P ∈ S dγ }.
d≥0
Kalai [62] gave a proof of Perles’ Skeleton Theorem for simplicial polytopes
that uses the Erdős-Rado Sunflower Lemma [40]. His proof implies a bound
of (γ + 1)k+1 (k + 1)! for MS (k, γ).
In the same paper he stated the following conjecture by Kalai, Kleinschmidt & Lee.
Conjecture 7.5.1 (Kalai, Kleinschmidt & Lee [62, Conjecture 2]).
Let (h0 , . . . , hd ) be the h-vector of a simplicial d-polytope, and let 1 ≤ k ≤
d − 1. For all simplicial d-polytopes with h-vector (h0 , . . . , hd ), the number
of empty k-simplices is maximized by the Billera-Lee polytopes constructed
in the proof of the sufficiency part of the g-Theorem [17] [18].
This conjecture was proven by Nagel [84, Theorem 2.3] following ideas by
Kalai [63, Theorem 20.5.35], who observed that Conjecture 7.5.1 essentially
follows from a result by Migliore & Nagel [81].
It was shown by Kalai [62, Theorem 3.8] that, for 0 ≤ j ≤ k and fixed
α, the number of empty k-simplices of a simplicial d-polytope with gj ≤ α
is bounded by a constant that is independent of d. Another proof of this
statement was given by Nagel [84, Theorem 4.20]. It is open whether such
a statement holds for general polytopes.
With respect to the number of empty simplices in simplicial polytopes,
Nagel proved the following optimal bounds for MS (k, γ).
Theorem 7.5.2 (Nagel [84, Theorem 4.15]). Let P be a simplicial dpolytope on d + γ + 1 vertices, γ ≥ 1 and k ≥ 1. Then
(
¡γ+k¢
,
if 1 ≤ k < d/2,
¡γ+⌊d/2⌋−1¢
M (k, P ) ≤ ¡γ+⌊d/2⌋¢ γ−1
,
if d/2 ≤ k < d.
+
γ−1
γ−1
Furthermore, the first bound is attained if P¡is the¢Billera-Lee d-polytope
, for 0 ≤ i ≤ m, and
with g-vector (g0 = γ, g1 , . . . , gm ), where gi = γ+i−1
i
m = min{k, ⌊d/2⌋}.
119
The bound for d/2 ≤ k is attained by the cyclic polytopes, that is,
the polytopes that arise as the convex hull of points on the moment curve,
according to the following result by Terai & Hibi [112].
Theorem 7.5.3 (Terai, Hibi [112]). Let Cd (n) be the cyclic d-polytope
on n = d + γ + 1 vertices. Then
µ
¶ µ
¶
γ + ⌊d/2⌋
γ + ⌊d/2⌋ − 1
M (k, Cd (n)) =
+
,
γ−1
γ−1
for every k ≥ d/2.
The following problem on empty simplices of simplicial polytopes was
posed by Kalai [62].
Problem 7.5.4 ([62, Problem 1]). Let C be a simplicial complex. Let
mk (C) denote the number of empty k-simplices of C.
Characterize the vectors (m1 , . . . , md−1 ) that arise from simplicial dpolytopes.
120
Chapter 8
Generalizations of
In this chapter we prove three generalizations of Perles’ Skeleton Theorem.
We prove a bound on the number of empty pyramids in strong PL
spheres parallel to the geometric proof given in Chapter 7. Finding equivalent topological statements for the geometric statements in Chapter 7 is
a large part of the work. In the end, we arrive at the same bound for the
number of empty pyramids.
We then pass to purely combinatorial objects.
It was remarked by Kalai, see [62, Section 2.4], that Perles had shown
his skeleton theorem for a class of lattices called pyramidally perfect [89].
As there is no written account of this generalization available we establish
this result in this chapter. In fact, we consider two classes of lattices: the
graded relatively complemented ones, and the pyramidally perfect ones.
The objects in this chapter are related to polytopes by the following
chain of inclusions:
{polytopal face lattices} ⊂ {face lattices of strong PL spheres}
⊂ {graded relatively complemented lattices}
⊂ {pyramidally perfect lattices}.
For d ≥ 4 there are many nonpolytopal strong PL (d − 1)-spheres; see
for example the Brückner sphere [33], or the ones constructed by Pfeifle &
Ziegler [91], and Kalai [60].
121
8. Generalizations of Perles’ Skeleton Theorem
The class of graded relatively complemented lattices is important, because it contains, besides the face lattices of polytopes, the class of geometric
lattices: the lattices of flats of matroids [85].
Recall that a finite graded lattice is geometric if it is relatively complemented and if it is semimodular, that is, the rank function rk satisfies for
any two elements x and y the inequality rk(x)+rk(y) ≥ rk(x ∨ y)+rk(x ∧ y).
It is easy to see that a polytopal face lattice is geometric if and only it
is boolean. (Show that, whenever a simplicial polytope is not isomorphic
to ∆d , there is a ridge and a vertex whose join is the polytope itself. The
statement then follows by induction.) The class of graded relatively complemented lattices is thus a much larger class than the class of face lattices
of polytopes. The same arguments apply for strong PL spheres.
We shall also see that the class of pyramidally perfect lattices is a larger
class than the class of graded relatively complemented ones. Thus the three
inclusions above are really proper inclusions.
Since we move from more concrete to more abstract objects, the reader
may be worried that we reprove things again and again. This is only true to
a small extent. In most cases the proofs for the more concrete objects yield
more information or better bounds. For example, the bounds on the size of
the kernel we prove mimic the above hierarchy. Although I do not know if
better bounds can be proved for the more general objects, the proofs given
definitely do not generalize. The two main reasons are that (a) there is
no equivalent to Grünbaums’s theorem on simplex refinements [50] (Theorem 2.1.2) for the two lattice classes under consideration, and that (b) the
class of pyramidally perfect lattices is not closed under taking upper intervals.
8.1
Strong PL Spheres
Definitions and notation are taken from [23, Chapter 4.7]. Basic references
for PL topology are Hudson [55], Zeeman [117], and Rourke & Sanderson [97].
8.1.1
Basic Notions
A geometric simplicial complex K is a finite collection of simplices in some
Re such that
(i) if F ∈ K then also all faces of F are in K,
(ii) if F, G ∈ K then F ∩ G is a face both of F and G.
122
8.1. Strong PL Spheres
Let K be a geometric simplicial complex in Re with vertices v1 , . . . , vn .
We express every point x of the underlying space || K || of K, that is, the
set of points in Re that lie in some simplexP
of K, by barycentric
coordinates.
P
These are real numbers λi such that x = ni=1 λi vi , ni=1 λi = 1, and such
that the set {vi : λi > 0} is the set ofP
vertices of a P
simplex of K. A map
f : ||P
K || → Rd is linear if f (x) = f ( ni=1 λi vi ) = ni=1 λi f (vi ) for every
x = ni=1 λi vi ∈ || K ||. It is piecewise linear if it is linear on some simplicial
subdivision of K.
Two geometric simplicial complexes K ⊂ Re and C ⊂ Rd are PL homeomorphic if there is a piecewise linear map f : || K || → || C || ⊂ Rd that is
a homeomophism.
Definition 8.1.1 (Simplicial PL ball, PL sphere). The underlying
space of a geometric simplicial complex is a PL d-ball (respectively, a PL
(d − 1)-sphere) if it admits a triangulation that is PL homeomorphic to the
standard d-simplex (respectively, boundary of the standard d-simplex).
Recall from Section 4.1 the definition of regular cell complexes and the
basic terminology connected with them.
Every regular d-dimensional cell complex P can be embedded with a
flat embedding of its order complex, which is isomorphic to the barycentric
subdivision. (A flat embedding or a geometric realization of an abstract
simplicial complex is a geometric simplicial complex that is combinatorially
isomorphic to the abstract simplicial complex.)
Definition 8.1.2 (PL ball, PL sphere; compare [23, Lemma 4.7.25]).
A regular cell complex P is a cellular PL d-ball (respectively, a cellular PL
(d − 1)-sphere) if one (and thus every) geometric realization of its order
complex is a simplicial PL d-ball (respectively, a simplicial PL (d − 1)sphere).
Clearly, this definition is equivalent to requiring the following: Every geometric realization of the order complex of P admits a triangulation that has
a simplicial subdivision that is combinatorially isomorphic to some simplicial subdivision of the d-simplex (respectively, boundary of the d-simplex).
Definition 8.1.3 (Strong PL sphere). A strongly regular cellular dsphere that is a PL d-sphere is called a strong PL d-sphere.
The structures defined in this section are related by inclusion as shown
in Figure 8.1.
If P is a strongly regular cellular (d−1)-sphere, we set γ(P ) := f0 −d−1.
It is easy to see that a strongly regular cellular (d − 1)-sphere has at least
123
regular
cell complexes
cellular
spheres
geometric
simplicial
complex
strongly regular
cellular sphere
cellular PL ball
PL ball
cellular PL sphere
PL sphere
strong PL sphere
Figure 8.1: Inclusion relations among the structures defined in this section. A
dotted line indicates the relation via barycentric subdivision.
d + 1 vertices. Thus γ(P ) ≥ 0 for every strongly regular cellular (d − 1)sphere.
b
The augmented face poset F(C)
of a regular cell complex C is the face
poset F(C), which has a 0̂ corresponding to the empty set by Definition 4.1.1, together with an element 1̂ such that F < 1̂ for every F ∈ F(C).
8.1.2
Basic Constructions and Properties
Let C be a regular cell complex. Then C × I, where I = [0, 1] denotes the
unit interval, is again a regular cell complex. The cells of C ×I are either
of type F × {0}, F × {1}, or they are of type F × I, where F is a cell of C.
The cell complex C ×I is called the prism over C.
The pyramid pyr(C) over C, also called the cone over C, is obtained from
C ×I by identifying all points in C ×{0}. The pyramid pyr(C) is again a
regular cell complex. The vertex that arises from identifying the points in
C ×{0} is the apex of pyr(C). The complex C is called the base of C. If
a complex C is a pyramid over a complex K with apex v, we also write
C = pyrv (K).
We need a number of properties of PL spheres that are collected in the
124
8.2. Empty Faces in Strong PL Spheres
following theorem.
Theorem 8.1.4 (cf. [23, Theorem 4.7.21] [23, Proposition 4.7.26]).
(i) (Newman’s theorem) The closure of the complement of a PL d-ball
embedded in a PL d-sphere is itself a PL d-ball.
(ii) The cone over a PL d-sphere is a PL (d + 1)-ball.
(iii) If P is a PL sphere, then every closed cell σ ∈ P is a PL ball.
(iv) If P is a PL sphere, then there exists a PL regular cell decomposition
P ∆ of the d-sphere with anti-isomorphic augmented face poset:
b ∆) ∼
b )op .
F(P
= F(P
Theorem 8.1.4 (iii) and (iv) are crucial for extending many of the arguments that work for polytopes to strong PL spheres. They allow us to
define vertex figures, or more general, face figures of PL spheres. Let F and
G be two cells of P with F ⊆ G. Then by the two latter properties of the
above theorem, the interval [F, G] in the face poset of P is the augmented
face poset of a strong PL sphere, denoted by G/F . Such a strong PL
sphere is called a quotient of P . The same is true for intervals [F, 1̂] in the
augmented poset of P , in which case we denote the resulting PL sphere by
P/F and call it a face figure of P . In the special case when F is a vertex,
we call P/F a vertex figure.
For our comfort in the combinatorial arguments that follow it is also
good to know that the augmented face poset of a strong PL sphere is
a graded lattice with diamond property (actually, this is true as well for
regular cell decompositions of spheres with intersection property; see [23,
Corollary 4.7.12]).
8.2
Empty Faces in Strong PL Spheres
In this section, we define empty faces in strong PL spheres and prove a
number of properties of them.
Let U ⊆ V(C) be a subset of the vertices of a regular cell complex C. The
subcomplex of C induced by U , denoted by C[U ], is the regular cell complex
induced by the vertex set U . That is, C[U ] consists precisely of all the cells
F of C with V(F ) ⊆ U . If a subcomplex is an induced subcomplex for some
set of vertices, we call it vertex-induced.
For k ≥ 1, an empty k-face of C is a vertex-induced subcomplex that is
a regular cell decomposition of the (k − 1)-sphere.
125
If this cell decomposition is combinatorially isomorphic to the boundary
of a pyramid or a simplex, the empty face is called empty k-pyramid or
empty k-simplex , respectively. If the dimension does not matter, we also
call the cell decomposition simply an empty face, empty pyramid or empty
simplex, respectively.
Empty 1-faces will also be called missing edges.
If C satisfies the intersection property, then clearly also C[U ] satisfies the
intersection property.
Let P be a strong PL sphere, M an empty face of P , and let F be a
cell of M . Let G1 , . . . , Gm be the cells of M that contain F , and let P/F
be a face figure of P at F . The strongly regular cell subcomplex of P/F
given by the set of cells {G1 /F, G2 /F, . . . , Gm /F }, where Gi /F is the cell
of P/F that corresponds to the cell Gi in P , is called the quotient of the
empty face M at F . We will denote this complex by M/F .
8.2.1
Empty Pyramids in Quotients
The following lemma is the topological equivalent to Lemma 7.1.3.
Lemma 8.2.1 ([23, Lemma 4.7.3]). Let S k be a k-dimensional sphere
and A ⊂ S k a proper subset of S k . Then A is not homeomorphic to a
k-sphere.
Proof. This follows, for example, from Alexander duality [82].
One of the key features of empty pyramids is that they behave nicely
with respect to intersection with cells and with other empty pyramids.
Lemma 8.2.2. Let P be a strongly regular cellular (d−1)-sphere, let F ⊂ P
be a k-cell of P , and let M be an empty k-pyramid of P . Then the strongly
regular cell complex F ∩ M is a cell of F and hence of P .
Proof. Since M is an empty k-pyramid, there is a (k − 1)-cell G ⊂ M and a
vertex v ∈ V(M ), such that M = ∂(pyrv (G)). By the intersection property,
the intersection F ′ := F ∩ G is a face of P .
If F ′ = G, then the vertex v is not a vertex of F . Otherwise, every
proper face of M is also a face of F . However, both ∂F and M are (k − 1)spheres, so ∂F = M , by Lemma 8.2.1. This contradicts the facts that F is
a k-face and that M is an empty k-face.
In the other case F ′ 6= G. Either the vertex v lies in F , in which case
the intersection F ∩ M equals pyrv (F ′ ), or v does not lie in F . In the latter
case we have F ∩ M = F ′ .
126
8.2. Empty Faces in Strong PL Spheres
(a) The two geometric 2-faces indicated
by the bold lines intersect in two components.
(b) The empty 2-face indicated by the
bold lines intersects the empty 2-face from
Figure 8.2(a) in two edges.
Figure 8.2: Empty faces in 3-polytopes.
The following lemma holds in much more generality for pyramidally
perfect lattices; see Lemma 8.9.5. We therefore postpone the proof until
later.
Lemma 8.2.3. Let P be a strongly regular cellular (d−1)-sphere, let k ≥ 1,
and let M1 and M2 be distinct empty k-pyramids of P . Then the regular
cell complex M1 ∩ M2 is a cell of P .
Remark 8.2.4. It is not true that the intersection of two geometric empty
faces in a polytope is a single face. Consider the geometric empty faces
of the regular 3-dimensional crosspolytope indicated by the bold lines in
Figure 8.2(a).
General empty faces can also intersect locally in a complex that is larger
than a single face. Consider, for example, the intersection of the empty 2face indicated by the bold lines in Figure 8.2(b) with the empty 2-face in
Figure 7.1 at the beginning of the previous chapter.
Lemma 8.2.5. Let k, d ∈ N with 1 ≤ k ≤ d − 1. Let P be a strong
PL (d − 1)-sphere, and let M = ∂ pyrv (G) be an empty k-pyramid, where
v ∈ V(P ) and G ⊂ P is a (k − 1)-cell of P .
Let F be a cell of M of dimension at most k − 2. Then M/F is an
empty (k − dim(F ) − 1)-face in P/F with M/F = ∂(pyrv (G)/F ).
Proof. Let G1 , . . . , Gm be the cells of M that contain the cell F . Denote by
P/F a face figure of P at F , that is, a strong PL sphere with face poset that
127
is isomorphic to the interval [F, 1̂] in the augmented face poset of P . For
i = 1, . . . , m, let G′i := Gi /F be the cells of M/F in P/F that correspond
to the faces G1 , . . . , Gm .
We claim that M/F is an empty cell. Suppose there is a (k −dim F −1)cell in P/F that has M/F as its boundary. Then this cell corresponds to
a k-cell F ′ in P that intersects M in G1 , . . . Gm . By Lemma 8.2.2, the set
of cells {G1 , . . . , Gm } then is exactly the set of cells of a cell H ⊂ P of
dimension at most k − 1. Since M is an empty k-pyramid and F has at
most dimension k − 2 there are at least two cells of dimension k − 1 that
contain F and that are cells of M . Consequently, they are containd in H.
This is clearly a contradiction.
Since the set {G1 , . . . , Gm } contains at least one cell of dimension k − 1,
the set of cells {G′1 , . . . , G′m } in P/F contains at least one ((k−1)−dim(F )−
1)-cell. Thus M/F is an empty (k − dim(F ) − 1)-cell.
Corollary 8.2.6. Let P be a strong (d − 1)-sphere and v ∈ V(P ). Let
2 ≤ k ≤ d − 1 and let M be an empty k-pyramid with v ∈ M . Then M/v is
an empty (k − 1)-pyramid of P/v if and only if at least one of the following
two cases applies:
(i) There is w ∈ V(P ) \ {v} and a (k − 1)-cell G with M = ∂ pyrw (G), or
(ii) there is a (k − 1)-pyramid G ⊂ P with M = ∂ pyrv (G).
Proof. We have that M/v = ∂ pyrw (G)/v for v ∈ V(P ), G is a (k − 1)-cell,
and that pyrw (G)/v is a pyramid if and only if v ∈ G or if G is a pyramid.
The statement then follows immediately from Lemma 8.2.5.
The following lemma is the equivalent of Corollary 7.1.5.
Lemma 8.2.7. Let P be a strong (d − 1)-sphere and let 1 ≤ k1 , k2 ≤ d − 1.
For i = 1, 2, let Mi be an empty ki -pyramid and suppose that M1 and
M2 are distinct. Let F be a cell of M1 and of M2 of dimension at most
min{k1 , k2 } − 2. Then M1 /F and M2 /F are distinct empty cells of P/F .
Proof. Let ki′ := ki − dim F − 1 for i = 1, 2. By Lemma 8.2.5, Mi /F is an
empty ki′ -cell of P/F for i = 1, 2. Clearly, if we have k1 6= k2 , the empty
cells M1 /F and M2 /F are distinct.
So assume k1 = k2 and let k := k1 = k2 . Consider the intersection
′ :=
F
M1 ∩ M2 . According to Lemma 8.2.3, F ′ is a cell of P of dimension
at most k − 1. We also have F ⊆ F ′ , since F was assumed to be a cell
of M1 and of M2 . Since Mi , i = 1, 2, is an empty k-pyramid there is at
least one (k − 1)-cell Gi of Mi distinct from F ′ that contains F , which was
128
8.3. Simplex Refinements of PL Spheres
assumed to be of dimension at most k − 2. Clearly Gi is not a cell of Mj
for {i, j} = {1, 2}, so Gi /F is not a cell of Mj /F . Thus M1 /F and M2 /F
are distinct empty cells.
Definition 8.2.8. Let P be a strong PL (d − 1)-sphere and let F a k-cell
of P . Denote by βF (P ) the number of empty (k + 1)-pyramids that contain
F.
Lemma 8.2.9 (cf. [62, Lemma 2.1]). Let P be a strong PL (d−1)-sphere
on d + γ + 1 vertices and F a cell of P . Then
βF (P ) ≤ γ(P ) − γ(P/F ) − γ(F ).
Proof. Suppose that dim(F ) = k. Then the quotient P/F has dim(P/F ) =
d − k − 1 and m := f0 (P/F ) is the number of (k + 1)-cells in P that contain
the cell F .
Let F1 , . . . , Fm be the (k + 1)-cells that contain F , and M1 , . . . , MβF (P )
the empty (k + 1)-pyramids that contain F .
For every 1 ≤ i ≤ m, let vi be a vertex in V(Fi )\ V(F ). Then vi 6= vj ,
whenever i 6= j, since Fi ∩ Fj = F . Likewise, for every 1 ≤ i ≤ βF (P ) let wi
be a vertex in V(Hi ) \ V(F ). Then also wi 6= wj whenever i 6= j, since by
Lemma 8.2.3 we have Mi ∩ Mj = F . Lemma 8.2.2 implies that Fi ∩ Mj = F
for all 1 ≤ i ≤ m and 1 ≤ j ≤ βF (P ), so we also have vi 6= wj .
This shows that
f0 (P/F ) + βF (P ) ≤ f0 (P ) − f0 (F ).
If we substitute using the equations f0 (P/F ) = d − k + γ(P/F ), f0 (P ) =
d + γ(P ) + 1, and f0 (F ) = k + γ(F ) + 1, we get
d − k + γ(P/F ) + βF (P ) ≤ d + γ(P ) + 1 − (k + γ(F ) + 1),
which, if simplified, is the statement we wanted to prove.
Corollary 8.2.10. Let P be a strong PL (d−1)-sphere on d+γ +1 vertices,
and let F be a cell of P . Then βF (P ) ≤ γ(P ) − γ(P/F ).
8.3
Simplex Refinements of PL Spheres
In this section we generalize Grünbaum’s theorem on simplex refinements
of polytopes [50], Theorem 2.1.2, to strong PL spheres.
We prove the generalization in two parts by (a) proving that the closed
star of a vertex is a refinement of the pyramid over the boundary of a vertex
figure at that vertex, and (b) using this property as a replacement for the
radial projection in the proof of Grünbaum’s theorem.
129
ast(v)
v
star(v)
(a) The star of the vertex v.
(b) The antistar of the vertex v.
Figure 8.3: The star and the antistar of the vertex of a 2-sphere (only partly
drawn). Their intersection is the link.
8.3.1
Stars, Antistars, and Links
We need to decompose a sphere into distinguished subcomplexes, the (closed)
star, the (closed) antistar, and the link. They are introduced in the following definition; compare with Definition 1.3.6 for polytopal complexes.
Definition 8.3.1 (Star, antistar, link). Let C be a regular cell comples,
and let F be a cell of C. We define
• the star at F by
starC (F ) := {G ∈ C : there is H ∈ C such that F ⊆ H and G ⊆ H},
• the antistar at F by astC (F ) := {G ∈ C : F 6⊆ G},
• and the link at F by linkC (F ) := astC (F ) ∩ starC (F );
see Figure 8.3.
A basic relation of these subcomplexes is given by the following lemma
that we need below.
Lemma 8.3.2. Let C be a regular cell complex and let F be a cell of C.
Then
[
linkC (F ) =
ast∂H (F ).
H∈ C,F ⊆H
130
8.3. Simplex Refinements of PL Spheres
Proof. We have
linkC (F ) = astC (F ) ∩ starC (F )
= {G ∈ C : there is H ∈ C : G ⊆ H, F ⊆ H, F 6⊆ G}
[
=
{G ∈ C : G ⊂ H, F 6⊆ G}
H∈C,F ⊆H
=
[
ast∂H (F ),
H∈C,F ⊆H
so the statement holds.
Lemma 8.3.3. Let P be a strong PL (d − 1)-sphere and v ∈ V(P ). Then
starP (v) and astP (v) are strong PL (d − 1)-balls, and linkP (v) is a strong
PL (d − 2)-sphere.
Proof. We prove the statement by induction on d. For dimensions d ≥ 3, every strong PL (d − 1)-sphere is isomorphic to the boundary of a d-polytope,
so the statement holds. (For d = 3, this is Steinitz’ theorem [108] [109];
see [118, Theorem 4.1].)
So let P be a strong PL (d − 1)-sphere, where d ≥ 4, and let v ∈ V(P ).
Form the strongly regular cell complex P̃ by replacing the star of P at
v be the pyramid over the link of P at v with apex ṽ. Then P and P̃
are PL homeomorphic. Indeed, by induction, the antistar of every facet
of P that contains v is a PL ball, so the pyramid over it is also a PL
ball. Thus, every facet F of P that contains v is PL homeomorphic to the
pyramid over the antistar of ∂F at ṽ. These homeomorphisms can be made
to coincide on intersections of facets (the technical statement one needs
here is that two simplicial subdivisions of the same underlying space have a
common refinement [117, Chapter 1]). Thus starP (v) and and starP̃ (ṽ) are
PL homeomorphic, and so are P and P̃ .
Since starP̃ (ṽ) is a cellular PL (d − 1)-ball (see [97, Chapter 2]), also
starP (v) is a cellular PL (d − 1)-ball. Newman’s theorem implies that also
astP (v) is a cellular PL (d − 1)-ball. Since the link is the boundary of the
star, it is a PL (d − 2)-sphere.
8.3.2
A Replacement for Radial Projection
Definition 8.3.4 (Refinement homeomorphism). Let C and K be regular cell complexes. A refinement homeomorphism is a homeomorphism
φ : k K k → k C k such that for every F ∈ K there is a subcomplex CF ⊆ C
with φ(kF k) = kCF k. We then say that C is a refinement of K.
131
The principal vertices of the refinement are the vertices φ(V(K)), that
is, the images of the vertices of K.
The following lemma replaces the radial projection in the proof of Theorem 2.1.2, Grünbaum’s theorem on simplex refinements of polytopes.
Lemma 8.3.5. Let P be a strong PL (d − 1)-sphere, and let v ∈ V(P ).
Furthermore, let P/v be a vertex figure of P at v.
Then the star starP (v) is a refinement of the pyramid over P/v such that
the principal vertices of the refinement are U ∪ {v}, where U is a subset of
the neighbors N (v) in the graph of P .
Proof. We denote by F/v the cell of P/v that correponds to the cell F of
P.
We inductively construct maps φi : skeli (pyrv (P/v)) → skeli (starP (v))
for i ∈ {0, 1, . . . , d − 1} that satisfy the following two properties:
(i) The map φi is a refinement homeomorphism onto its image.
(ii) For every j-cell F of P that contains v with j ≤ min{i + 1, d − 1} we
have that φi (∂ pyrv (F/v)) = ∂F and φi (F/v) = astF (v).
Clearly, the map φ0 exists, so suppose that 1 ≤ 1 ≤ d − 2.
Suppose that G/v is an (i−1)-cell of P/v that corresponds to the i-cell G
of P . We define φi on pyrv (G/v) as follows. By (ii), the map φi−1 is defined
on the boundary of pyrv (G/v). We define φi identically on the boundary
and extend it on the interior to a homeomorphism between pyrv (G/v) and
the cell G. (It is easy to prove that this can be done; see for example [23,
Lemma 4.7.2].)
Next, suppose that G/v is an i-cell of P/v that corresponds to the (i+1)cell G of P . By (ii) and by Lemma 8.3.2, we have that φi−1 (∂G/v) =
link∂G (F ). Define φi identically to φi−1 on ∂G/v. Then extend this to
a homeomorphism between G/v and ast∂G (v), which is a PL i-ball by
Lemma 8.3.3.
The map φi is a refinement homeomorphism from skeli (pyrv (P/v)) to
skeli (starP (v)), because P has the intersection property. By construction it
is clear that the map satisfies (ii).
By Lemma 8.3.2 and by (ii) we have that the map φd−1 restricted to the
(d − 2)-skeleton of P/v is a refinement homeomorphism between P/v and
linkP (v), as both are (d − 2)-spheres and consequently
skeld−2 (P/v) = P/v
and
skeld−2 (linkP (v)) = linkP (v).
Thus φd−1 is a refinement homeomorphism between pyrv (P/v) and starP (v).
By (ii) for j = 1 we have that the principal vertices are U ∪ {v}, where U
is a subset of the neighbors of v.
132
8.4. Perles’ Skeleton Theorem for PL Spheres
8.3.3
Proof of Grünbaum’s Theorem for Strong PL Spheres
Definition 8.3.6 (Rooted refinement). Let P be a strong PL (d − 1)sphere and v a vertex of P . We say that a refinement homeomorphism with
principal vertices U ∪ {v} is rooted at v if U is a subset of the neighbors of
v in G(P ).
Theorem 8.3.7. Let P be a strong PL (d − 1)-sphere, and let v be a vertex
of P .
Then there is a refinement homeomorphism from ∂∆d to P that is rooted
at v.
Proof. We prove the statement by induction on d. For d ≤ 1, every (d − 1)sphere is isomorphic to the boundary of ∆d .
Let d ≥ 2 and let P/v be a vertex figure of P at v. Let v0 be a vertex
of ∆d . By induction there is a refinement homeomorphism
φv : link∂∆d (v0 ) = ∂∆d−1 → P/v,
so clearly there is a refinement homeomorphism φ∗v from pyrv0 (link∂∆d (v0 )) =
star∂∆d (v0 ) to pyrv (P/v). By Lemma 8.3.5, starP (v) is a refinement of
pyrv (P/v). Concatenating the refinement φ∗v with this refinement, we have
a refinement homeomorphism from star∂∆d (v0 ) to starP (v). Extending this
homeomorphism onto ast∂∆d (v0 ) yields the desired refinement homeomorphism. The statement about the principal vertices follows from the corresponding statement in Lemma 8.3.5.
Lemma 8.3.5 yields in principle the same as radial projection. In particular, with a more detailed analysis one could prove the strengthenings that
we proved in Theorem 2.1.2 for polytopes analogously for PL spheres. As
we have no use for them, we contain ourselves with the version at hand.
Corollary 8.3.8 (cf. Corollary 2.1.5). Let P be a strong PL (d − 1)sphere, and let v ∈ V(P ) a vertex of P . Then G(P ) contains a subdivision
of Kd+1 rooted at v.
Corollary 8.3.8 was proved in much more generality by Barnette [8] [10];
but see also the discussion in Section 4.3.
8.4
Perles’ Skeleton Theorem for PL Spheres
We replace the geometric proof of the bound on the number of missing
edges by an argument that involves the subdivision of a complete graph.
133
Theorem 8.4.1. Let P be a strong PL (d − 1)-sphere on d + γ + 1 vertices.
Then the number of missing edges is bounded by γ(γ + 1).
Proof. Let v be a vertex of P and let G be a refinement of Kd+1 rooted at
v. Let D denote the set of principal vertices of G and B = V(P ) \ D.
If for vertices v1 , v2 ∈ D the edge v1 v2 is a missing edge of P , then at
least one of the vertices in B subdivides this edge. Therefore there can be
at most γ missing edges between vertices in D. Additionally, every vertex
in B has degree at least d in P , thus there can be at most γ 2 empty edges
incident to B. This totals to at most γ(γ + 1) empty edges.
The following two lemmas yield the generalization of Lemma 7.2.3. We
had proven this lemma for polytopes with the aid of a refinement homeomorphism with affinely independent principal vertices. The following statement
is the combinatorial analogon to this.
Lemma 8.4.2. Let P be a strong PL (d − 1)-sphere and let φ : ∂∆d → P
be a refinement homeomorphism with principal vertices D ⊆ V(P ). Let
1 ≤ k ≤ d and let U ⊆ D with |U | = k.
If U is the vertex-set of a face F , that is, U = V(F ), then F is a
(k − 1)-simplex.
Proof. Let j := dim F . By the definition of a refinement homeomorphism,
there is a j-simplex τ ⊆ ∂∆d such that F is a face of the strongly regular
cell complex φ(τ ).
Let τ ′ ⊆ ∆d be the (k − 1)-simplex with vertices φ−1 (U ) = φ−1 (V(F )).
Since j ≤ k − 1 we have that τ ′ and τ intersect in the set of vertices
φ−1 (V(F )). Thus we have τ ′ = τ and it follows that dim F = k − 1. So F
is a (k − 1)-simplex.
The previous lemma yields that every empty face induced by a subset
of principal vertices is simplicial. Of course, this implies that an empty
pyramid induced by a subset of principal vertices is an empty simplex.
Lemma 8.4.3. Let P be a strong PL (d−1)-sphere on d+γ +1 vertices and
let φ : ∂∆d → P be a refinement homeomorphism with principal vertices
D ⊆ V(P ).
Then there are at most γ empty pyramids induced by vertices in D and
they are all empty simplices.
Proof. If M is an empty ℓ-pyramid induced by vertices in D, that is,
V(M ) ⊆ D, then by Lemma 8.4.2 every proper face of M is an (ℓ − 1)simplex. Thus M is an empty ℓ-simplex on ℓ + 1 vertices.
134
8.4. Perles’ Skeleton Theorem for PL Spheres
Let τ ⊆ ∂∆d be the ℓ-simplex with V(τ ) = φ−1 (V(M )). The strongly
regular complex φ(τ ) ⊆ P is a pure ℓ-dimensional complex. Therefore,
φ(τ ) contains a vertex besides the vertices of M . Since φ is a bijection, we
then have an injection from the set of empty pyramids induced on D into
B := V(P ) \ D.
We conclude that D can induce at most |B| = γ empty pyramids, which
are necessarily all empty simplices.
We now come to the proof of the main theorem of this section, the
generalization of Theorem 7.2.4. The argument is slightly more intricate as
empty pyramids are not necessarily empty pyramids in vertex figures. We
have to make sure that we nevertheless count all of them.
Theorem 8.4.4. Let P be a strong PL (d − 1)-sphere on d + γ + 1 vertices
and 1 ≤ k ≤ d − 1. Then the total number of empty ℓ-pyramids with ℓ ≤ k
is bounded by γ(γ + 1)k .
Proof. We prove the statement by induction on k. Let us fix some vertex
v ∈ V(P ), and consider a refinement homeomorphism φ : ∂∆d → P rooted
at v. Let the principal vertices be D = φ(V(∆d )), and B = V(P ) \ D. Then
the size of B is |B| = γ.
Theorem 8.4.1 yields that the number of missing edges is bounded by
γ(γ + 1).
So, let k ≥ 2. Lemma 8.4.3 implies that the number of empty pyramids
induced by vertices in D is bounded by γ.
The remaining empty ℓ-pyramids, ℓ ≤ k, of P contain each at least one
vertex in B. Let M be such an empty ℓ-pyramid with ℓ ≥ 2.
Claim. There is w ∈ V(P ) and an (ℓ − 1)-cell F with M = ∂ pyrw (F ) and
V(F ) ∩ B 6= ∅.
Proof of the claim. Suppose otherwise that for all possible choices of w and
F with M = ∂ pyrw (F ) we have V(F ) ∩ B = ∅. Fix one such choice
w and F . Since all vertices of F lie in D we get by Lemma 8.4.3
that F is an (ℓ − 1)-simplex. Let F ′ ⊆ F be a facet of F and w̃ ∈
V(F ) \ V(F ′ ). Then for F̃ := pyrw (F ′ ) and w̃ we have M = pyrw̃ (F̃ )
and V(F̃ ) ∩ B 6= ∅, since w ∈ V(F̃ ) ∩ B. This proves the claim.
Corollary 8.2.6 now implies that for every empty ℓ-pyramid with 2 ≤
ℓ ≤ k that is not induced by vertices in D there is a vertex b ∈ B such that
M/b is an empty (ℓ − 1)-pyramid of P/b.
Let
Γ(b) := {M ′ : there is an empty ℓ-pyramid, ℓ ≤ k, in P with M/b = M ′ },
135
and
Γ :=
[
Γ(b)
b∈B
By Lemma 8.2.7 we have a surjection from Γ to the set of empty ℓ-pyramids,
2 ≤ ℓ ≤ k, of P that contain a vertex of B. This shows that the number of
empty ℓ-pyramids with 2 ≤ ℓ ≤ k of P is bounded by γ + |Γ|.
For b ∈ B, let P/b be a vertex figure. The boundary of P/b is a strong
PL (d − 2)-sphere on (d − 1) + γ ′ + 1 = d + γ ′ vertices, where γ ′ = γ −
βw (P ). By the induction hypothesis, there are at most γ ′ (γ ′ + 1)k−1 empty
ℓ-pyramids in P/b with ℓ ≤ k − 1. Additionally, there are at most βb (P )
many missing edges at b. Therefore, we count at most γ ′ (γ ′ +1)k−1 +βb (P ) ≤
γ(γ + 1)k−1 empty pyramids of P at b. Thus, the total number of empty
ℓ-pyramids with ℓ ≤ k is bounded by
γ + γ 2 (γ + 1)k−1 ≤ γ(γ + 1)k ,
as claimed.
8.5
Pyramidally Perfect Lattices
According to Kalai [62], Perles’ original proof of his theorem in [89] applies
to pyramidally perfect lattices. Here is the definition of this class of lattices.
Definition 8.5.1 (Pyramidally perfect lattice; see [62]). Let L be a
graded atomic lattice, a ∈ A(L), and x ∈ L with a 6≤ x. Then a is pyramidal
over x if A(x ∨ a) = A(x) ∪ {a}.
The lattice L is pyramidally perfect if every atom that is pyramidal over
x ∈ L is also pyramidal over every y ≤ x.
A complete list of pyramidally perfect lattices with at most 8 elements
is given in Figure 8.4.
The following proposition is immediate from the definitions of graded,
atomic, and pyramidally perfect. It is however important to note (a) the
usefulness of this proposition, as it allows us to do induction by taking
lower intervals, and (b) that the condition that x equals 0̂ in (ii) and (iii)
cannot be dropped. For example, the rightmost lattice in Figure 8.4(b) is
the smallest example of a pyramidally perfect lattice L that has nonatomic
upper intervals.
Proposition 8.5.2. Let L be a lattice, x, y ∈ L, and let L′ := [x, y].
(i) If L is graded, then L′ is graded.
136
8.6. Boolean Intervals
a
(a) All pyramidally perfect lattices on at
most 7 elements.
b
(b) All pyramidally perfect lattices on 8
elements. The rightmost lattice has upper intervals [a, 1̂] and [b, 1̂] that are not
atomic.
Figure 8.4: A complete list of pyramidally perfect lattices on at most 8 elements.
(ii) If L is atomic and x = 0̂, then L′ is atomic.
(iii) If L is pyramidally perfect and x = 0̂, then L′ is pyramidally perfect.
8.6
Boolean Intervals
Definition 8.6.1 (Direct product [107]). Let P1 and P2 be two posets.
The product of P1 and P2 is the poset on the set P1 × P2 in which (x1 , x2 ) ≤
(y1 , y2 ) if and only if x1 ≤ y1 and x2 ≤ y2 .
If L1 and L2 are lattices then L1 × L2 is a lattice. The direct product
of two polytopal lattices is a lattice that is isomorphic to the face lattice of
the join of the two corresponding polytopes.
Definition 8.6.2 (Product with a finite set). Let L be a lattice, and
let A be a finite set with A ∩ L = ∅. We define the product of L with A by
L ∗ A := L × B(A).
If for x ∈ L and A ⊆ A(L) \ A(x) there is y ∈ L such that x ≤ y,
[0̂, y] ∼
= [0̂, x] ∗ A, and A(y) = A(x) ∪ A, we also write y = x ∗ A.
If A = {a} we write pyra (L) := L ∗ A and call pyra (L) a pyramid over
base L with apex a. If for x ∈ L and a ∈ A(L) \ A(x) there is y ∈ L
such that x ≤ y, [0̂, y] ∼
= pyra ([0̂, x]), and A(y) = A(x) ∪ {a}, we also write
y = pyra (x).
137
If L is a graded lattice of rank r and n := |A|, then L ∗ A has rank r + n.
Definition 8.6.3 (γ). For a graded atomic lattice of rank r on n atoms we
define γ(L) := n − r.
The parameter γ behaves as for polytopes, as the following lemma shows.
Lemma 8.6.4. Let L be a graded atomic lattice.
(i) For every lower interval of L we have γ(L) ≥ γ(L′ ). In particular,
γ(L) ≥ 0.
(ii) If γ(L) = γ(L′ ) for a lower interval such that rk(L′ ) = rk(L) − 1, then
a ∈ A(L) \ A(L′ ) is pyramidal over L′ .
Proof. Let L be a graded atomic lattice of rank r := rk(L). We prove the
statement by induction on r, where the case r = 0 is trivial.
Suppose that r ≥ 1 and let n := | A(L)|. Let x ∈ L be of rank r − 1.
Let L′ := [0̂, x] and n′ := | A(L′ )|.
By induction, γ(L′ ) = γ(L′′ ) for every lower interval L′′ in L′ . Since
L is atomic there is at least one atom in A(L) with a 6≤ x, as otherwise
1̂ would not be the join of atoms. Thus n ≥ n′ + 1 and it follows that
n − r ≥ n′ + 1 − r = n′ − (r − 1) = γ(L′ ). Thus we have γ(L) ≥ γ(L′ ) for
every lower interval L′ in L.
If γ(L) = γ(L′ ) for L′ with rk(L′ ) = r − 1, there is exactly one atom
a ∈ A(L) \ A(L′ ), that is, a is pyramidal over x.
For pyramidally perfect lattices this lemma immediately implies the following corollary; compare [23, Exercise 4.4(b)].
Corollary 8.6.5. Let L be a pyramidally perfect lattice. If γ(L) = γ(L′ )
for a lower interval L′ with rk(L′ ) = rk(L) − 1 then L is a pyramid over L′ .
In particular, if γ(L) = 0, then the lattice L is boolean.
We now formulate and prove the analog of Lemma 5.4.2. This lemma
stated that a d-polytope on d + γ + 1 vertices has a (d − γ)-face that is a
simplex.
Lemma 8.6.6. Let L be a pyramidally perfect lattice of rank r on r + γ
atoms and x ∈ L with rk(x) ≤ r − γ.
Then there is y ∈ L of rank rk(y) = r − γ and a set of atoms A ⊆
A(L) with A ∩ A(x) = ∅, such that x ∗ A = y. The set A is given by
A = A(y) \ A(x).
138
8.7. Reconstruction of Skeleta
Proof. We prove the statement by induction on r. The case r = 0 is trivial,
so suppose that the statement is true for every pyramidally perfect lattice
of rank r − 1, and let x be an element of L.
If rk(x) = r − γ, take y := x and A := ∅ and the statement holds.
If rk(x) < r − γ we have in particular that x 6= 1̂. Let x′ be a coatom
of L with x ∈ L′ := [0̂, x′ ]. Such a coatom exists, since L is graded. Let
γ ′ = γ(L′ ).
We have rk(L′ ) = r − 1 and 0 ≤ γ ′ ≤ γ by Lemma 8.6.4. By the
induction hypothesis, there is y ′ ∈ L′ of rank r − 1 − γ ′ and a set of atoms
A′ = A(x) \ A(y ′ ), such that x ∗ A′ = y ′ .
If γ ′ ≤ γ − 1, then we set y := y ′ and A := A′ and the statement follows.
Otherwise, L is a pyramid over L′ by Corollary 8.6.5. Let a ∈ A(L) \ A(L′ ).
Then there is y ∈ L with y = pyra (y ′ ). With A := A′ ∪ {a} we then get the
statement.
Corollary 8.6.7. Every atom of a pyramidally perfect lattice of rank r on
r + γ atoms is contained in a boolean interval of rank r − γ.
8.7
Reconstruction of Skeleta
Definition 8.7.1 (Induced sets). Let L be a pyramidally perfect lattice
and let A ⊆ A(L). We denote by L[A] the poset on the set
L[A] := {x ∈ L : A(x) ⊆ A},
with the order induced by L. We call L[A] the subset of L induced by A. If
L′ ⊆ L is induced by some set of atoms, we say that L′ is an atom-induced
subset of L.
An atom-induced subset of a lattice is a finite meet-semilattice with 0̂.
If L is graded, then L′ is graded, too.
Definition 8.7.2 (k-skeleton). The k-skeleton of a graded poset L is the
set of all elements of rank at most k. We denote the k-skeleton by skelk (L).
Definition 8.7.3 (Kernel). Let L be a pyramidally perfect lattice and let
k be an integer with k ≥ 1. Let
A := {a ∈ A(L) : there is x ∈ L such that rk(x) ≤ k − 1
and a is not pyramidal over x}.
We define the k-kernel Kerk (L) of L to be the k-skeleton of the subset of
L induced by A, that is, Kerk (L) := L[A].
139
On the face poset level, the Definition 6.1.4 given for the kernel of polytopes coincides with the above definition of the kernel.
To show this, we have to show that the set A defined in Definition 8.7.3,
considered as a set of vertices of a polytope P , is exactly the set of vertices
in empty pyramids of P of dimension at most k, that is, A = V(Kerk (P )).
If a ∈ A, let j be the smallest dimension such that a is not pyramidal
over every face of dimension j. By the definition of A, there is at least one
face of dimension at most k − 1 such that a is not pyramidal over this face.
This implies that 0 ≤ j ≤ k − 1. Let F be a j-face, over which a is not
pyramidal. Then, by the choice of j, the vertex a is the apex of an empty
pyramid with base F , and thus lies in an empty pyramid of dimension at
most k.
If a ∈ V(Kerk (P )), then either a is an apex of an empty ℓ-pyramid with
ℓ ≤ k, and so it is not pyramidal over some (ℓ − 1)-face, or it is the vertex
of a base F of an empty pyramid M . In this case, let G ⊂ F be a facet of
F that does not contain a, and let v ∈ V(P ) such that M = ∂(pyrv (F )).
Then pyrv (G) is a (k − 1)-face over which a is not pyramidal.
This equivalence of the definitions also follows from Corollary 8.9.4 that
we prove below.
We now prove the generalization of Lemma 6.1.5, which stated that the
k-skeleton of a polytope is reconstructable from the k-kernel.
Lemma 8.7.4. Let L be a pyramidally perfect lattice. Then the k-skeleton
of L can be reconstructed, up to isomorphism, from Kerk (L) and the total
number of atoms.
Proof. Given Kerk (L) and A, we define a poset on the set
K := {(x, B) : x ∈ Kerk (L), B ⊆ A and rk(x) + |B| ≤ k},
by the following order: If y1 = (x1 , B1 ) and y2 = (x2 , B2 ) are two elements
of K with x1 , x2 ∈ Kerk (L) and B1 , B2 ⊆ A, then y1 ≤K y2 if and only if
x1 ≤ x2 and B1 ⊆ B2 .
Then there is an order-preserving injective map from K to skelk (L).
Let y ∈ K with y = (x, B). As B ∩ A(Kerk (L)) = ∅, every atom in B is
pyramidal over every element of rank at most k − 1. But then there is a
unique element y ′ with A(y ′ ) = A(x) ∪ B and y ′ = x ∗ B. Clearly, the map
π : K → skelk (L) defined by π(y) = y ′ is order-preserving.
But there is also an order-preserving injective map from skelk (L) to K.
Let y ∈ skelk (L). Let A′ = A(y) \ A and B = A(y) ∩ A, that is, A′ , B is a
partition of the atoms of y into elements that lie in the k-kernel and those
that do not.
140
8.8. Relatively Complemented Lattices
Let
x :=
_
a′ ∈A′
a′ .
If we can show that A(x) = A′ , then it follows that y ′ := (x, B) is in K and
that the map π ∗ : skelk (L) → K defined by π ∗ (y) = y ′ is injective.
Assume to the contrary that there is b ∈ B with b ∈ A(x). Let A′′ ⊆ A′
be the smallest subset of A′ such that b ∈ A(x′′ ), where
x′′ :=
_
a′′ ∈A′′
a′′ .
Clearly, A′′ 6= ∅. Let a′′ ∈ A′′ , let Z = A′′ \ {a′′ }, and denote by z the join
of the atoms in Z. Then clearly
A(z) 6= A′′ ,
b∈
/ A(z)
and
z ∨ b = x′′ .
But then b is not pyramidal over z, which is of rank rk(z) ≤ k − 1. This is
a contradiction to b ∈ B ⊆ A.
8.8
Relatively Complemented Lattices
Recall that a lattice L is complemented if every element x ∈ L has a
complementary element y ∈ L, that is, an element y such that x ∨ y = 1̂
and x ∧ y = 0̂, and that it is relatively complemented if every interval is
complemented.
Lemma 8.8.1. Let L be a graded relatively complemented lattice. Then L
is pyramidally perfect.
Proof. The lattice L is atomic and graded by Theorem 1.2.2, so it only
remains to show that whenever an atom is pyramidal over some element, it
is also pyramidal over every element below that element. We prove this by
induction on the rank of L, where the base case is trivial.
Suppose that rk(L) ≥ 1. Let a ∈ A(L) and suppose that a is pyramidal
over x for some x with a 6≤ x. We can assume that x is a coatom of
L, otherwise we apply the induction hypothesis on the lattice [0̂, x ∨ a] ∼
=
pyra ([0̂, x]) and get that a is pyramidal over every y ≤ x. Furthermore, we
only need to show that a is pyramidal over every element covered by x. By
induction the statement then follows.
Let y be one of the elements in L that are covered by x. Then rk(y) =
r − 2. By Theorem 1.2.2, the interval [y, 1̂] contains at least 4 elements,
that is, it contains at least one element of rank r − 1 besides x.
141
Claim. There is exactly one element in [y, 1̂] of rank r − 1 besides x, and
this element is y ∨ a. Furthermore, A(y ∨ a) = A(y) ∪ {a}.
Proof of the claim. Since a is pyramidal over x, we have
·
A(L) = A(x)∪{a}.
As L is atomic, an element z of [y, 1̂] that is not x has to satisfy
A(z) = A(y) ∪ {a}. Otherwise, A(z) contains a proper superset of
atoms of y whose join is x. But then
_
a′ = 1̂,
′
a ∈A(z)
in contradiction to rk(z) = r − 1. Clearly, we then have z = y ∨ a,
and the claim is proved.
y.
Thus, we have shown that A(y ∨ a) = A(y) ∪ {a}, so a is pyramidal over
The list of pyramidally perfect lattices in Figure 8.4 shows that every
pyramidally perfect lattice of rank at most 2 is a graded relatively complemented lattice. However, for rank larger than 2 this is false, as the rightmost
example in Figure 8.4(b) shows.
8.9
Empty Pyramids in Lattices
We generalize the notion of “empty pyramid” to pyramidally perfect lattices. We also prove some of the lemmas of Chapter 7 in this generality.
At some point, when we start to take upper intervals, we have to restrict
the proofs to graded relatively complemented lattices. Indeed, the largest
subclass of pyramidally perfect lattices that is closed under taking upper
intervals is the class of graded relatively complemented lattices.
We must be careful when generalizing empty pyramids to pyramidally
perfect lattices, as we still want to be able to reconstruct the k-skeleton
from the set of vertices in empty pyramids.
One could say that the lattice in Figure 8.5(c), which is the lattice of
flats of the geometry in Figure 8.5(a), does not have any “missing edges,”
as any two atoms do have an element of rank 2 that lies above both of them.
Yet we need to be able to distinguish the 2-skeleton of this lattice from the
2-skeleton of the lattice in Figure 8.5(d), which is the lattice of flats of the
geometry in Figure 8.5(b).
As we show in Corollary 8.9.4, the following definition of empty pyramids
satisfies this requirement.
142
8.9. Empty Pyramids in Lattices
(a) A geometry on 4 points . . .
(b) Another geometry on 4 points . . .
(c) . . . and its lattice of flats.
(d) . . . and its lattice of flats.
Figure 8.5: Two geometries of rank 3 on 4 points with nonisomorphic 2-skeleta.
Definition 8.9.1 (Empty pyramid). Let L be a pyramidally perfect lattice, M ⊆ L an atom-induced subset of L, and
k := max{rk(x) : x ∈ M } + 1.
Suppose there is an element y ∈ M and an atom a ∈ A(M ) with
(i) rk(y) = k − 1,
(ii) {a} = A(M ) \ A(y),
(iii) a is pyramidal over every x < y, and
(iv) a is not pyramidal (in L) over y.
Then we call M an empty k-pyramid.
M , respectively.
We call y a base and a an apex of
Lemma 8.9.2. Let L be a pyramidally perfect lattice and k ≥ 2. Let M be
an empty k-pyramid, and let y ∈ M and a ∈ A(M ) be a base and apex of
M , respectively.
Then every element of M of rank k − 1 is either y or it can be written
as z ∨ a with z ⋖ y.
143
Proof. Let ỹ 6= y be of rank k − 1. Then a ∈ A(ỹ), otherwise A(ỹ) ⊆ A(y)
and ỹ = y. Let
_
b.
z :=
b∈A(ỹ)\{a}
Then z < y, since A(ỹ) \ {a} ⊆ A(y). Then a is pyramidal over z and
a ∨ z = ỹ. This implies that z ⋖ y.
It follows that, if M is an empty k-pyramid, the poset M̂ := M ∪ {1̂},
where 1̂ is an additional element with x < 1̂ for all x ∈ M , is isomorphic to
pyra (y) and thus that M̂ is a pyramidally perfect lattice.
Lemma 8.9.3. Let L be a pyramidally perfect lattice and k ≥ 2. Let M
be an empty k-pyramid and a ∈ A(M ). Then there is an empty ℓ-pyramid
M ′ ⊆ M with ℓ ≤ k such that a is an apex of M ′ .
Proof. Suppose there is no y in M with rk(y) ≤ k − 2 such that a is
pyramidal over every x < y but not over y, that is, there is no empty ℓpyramid M ′ ⊆ M with ℓ ≤ k − 1. Since a is pyramidal over 0̂, we get, by
induction, that a is pyramidal over every element of rank k − 2.
Since M̂ = M ∪ {1̂} is atomic, there is y ∈ M of rank k − 1 such that
a∈
/ A(y). It remains only to show that {a} = A(M ) \ A(y).
Let z ∈ M and b ∈ M with M̂ = pyrb (z). Let ỹ be covered by y and
b∈
/ A(ỹ) (such a ỹ exists since [0̂, y] is atomic). Now a is pyramidal over ỹ,
that is,
A(ỹ ∨ a) = A(ỹ) ∪ {a} 6∋ b.
By Lemma 8.9.2 we must have ỹ ∨ a = z. But then
A(ỹ ∨ a) = A(z) = A(M ) \ {b}
and we have A(ỹ) = A(M ) \ {a, b} and ỹ ∨ b = y, that is, {a} = A(M ) \
A(y).
This implies that the set of atoms in the k-kernel of L equals the set of
atoms in empty ℓ-pyramids with ℓ ≤ k.
Corollary 8.9.4. Let L be a pyramidally perfect lattice and k ≥ 1. Let
A = A(Kerk (L)) and B be the set of atoms in empty ℓ-pyramids with ℓ ≤ k.
Then A = B.
Proof. This follows at once from Lemma 8.9.3, as the case k = 1 is trivial.
The following lemma is the generalization of Lemma 8.2.3 to pyramidally
perfect lattices.
144
8.10. Empty Pyramids in Upper Intervals
Lemma 8.9.5. Let L be a pyramidally perfect lattice. Let M1 and M2 be
empty k-pyramids of L such that M1 6= M2 . Then M1 ∩ M2 is an interval
in L, that is, there is y ∈ L with [0̂, y] = M1 ∩ M2 .
Proof. Since, for i = 1, 2, Mi is an empty k-pyramid, there are ai ∈ A(Mi )
and xi ∈ Mi with rk(xi ) = k − 1, such that Mi ∪ {1̂} = pyrai (xi ).
Let ỹ := x1 ∧ x2 and M̃ = M1 ∩ M2 ⊂ L. We distinguish three cases.
Case (i). Suppose that a1 ∈
/ M2 and a2 ∈
/ M1 . Then M̃ = [0̂, x1 ] ∩ [0̂, x2 ],
and the statement holds for y := ỹ.
Case (ii). Suppose that, up to symmetry, a1 ∈ M2 and a2 ∈
/ M1 .
If x1 = ỹ = x1 ∧ x2 , then x1 = x2 , because rk(x1 ) = rk(x2 ). Thus
A(M1 ) = A(x1 ) ∪ {a1 } = A(M2 ) \ {a2 } = A(x2 ). This is not possible, as
M1 must have at least one element of rank k − 1 besides x1 = x2 . Hence,
me must have x1 6= x1 ∧ x2 . In this case, a1 is pyramidal over x1 ∧ x2 . If we
set y := a1 ∨(x1 ∧ x2 ), then clearly M̃ = [0̂, y].
Case (iii). In the final case we have a1 ∈ M2 and a2 ∈ M1 .
If x1 = x1 ∧ x2 = x2 , then clearly a1 = a2 and thus M1 = M2 . Also, if
a1 = a2 and x1 6= x2 then a1 = a2 is pyramidal over (x1 ∧ x2 ), and we set
y := a1 ∨(x1 ∧ x2 ).
So, we can assume that x1 6= x2 and a1 6= a2 . We have a1 ∨(x1 ∧ x2 ) < x2
or a2 ∨(x1 ∧ x2 ) < x1 , since otherwise M1 ⊆ M2 and M2 ⊆ M1 . Because
both a1 and a2 are pyramidal over x1 ∧ x2 the elements a1 ∨(x1 ∧ x2 ) and
a2 ∨(x1 ∧ x2 ) have the same rank. Thus a1 ∨(x1 ∧ x2 ) < x2 and a2 ∨(x1 ∧ x2 ) <
x1 . Then a2 is pyramidal over a1 ∨(x1 ∧ x2 ) and we get the statement with
y := a2 ∨ a1 ∨(x1 ∧ x2 ).
8.10
Empty Pyramids in Upper Intervals
In this section, the proofs and statements are merely valid for graded relatively complemented lattices, as the upper intervals in a pyramidally perfect
lattice are not necessarily pyramidally perfect.
Lemma 8.10.1. Let L be a graded relatively complemented lattice, let M1
be an empty k1 -pyramid and M2 be an empty k2 -pyramid distinct from M1 ,
and x ∈ M1 ∩ M2 . Suppose that the sets M1′ := {y : y ≥ x and y ∈ M1 }
and M2′ := {y : y ≥ x and y ∈ M2 } are empty pyramids in [x, 1̂]. Then
M1′ 6= M2′ .
Proof. The statement is clearly true if k1 6= k2 . If k1 = k2 we have by
Lemma 8.9.5 that there is y ∈ L with [0̂, y] = M1 ∩ M2 , since M1 6= M2 .
145
a
b
c
(a)
(b)
Figure 8.6: Two types of empty pyramids that do not occur in polytopal face
lattices.
Since x ∈ M1 ∩M2 we have x ≤ y and thus [x, y] = M1′ ∩M2′ ⊆ [x, 1̂]. We
had assumed that M1′ and M2′ are empty pyramids in [x, 1̂]. Accordingly,
we have M1′ 6= M1′ ∩ M2′ and M2′ 6= M1′ ∩ M2′ . This implies M1′ 6= M2′ .
The next lemma is probably the most technical statement we need for
the proof of Perles’ Skeleton Theorem for graded relatively complemented
lattices. It generalizes Lemma 8.2.9 for polytopes, as it bounds the number
of empty pyramids that “disappear” in upper intervals.
The proof is considerably more complicated than for polytopes. This
does not only have technical reasons. The situation for graded relatively
complemented lattices is really more complicated than for polytopes.
Here are two examples of situations we have to cope with that do not
appear in polytopal face lattices. The bold part of Figure 8.6(a) is an empty
2-pyramid, because A(a ∨ b) = {a, b, c}. However, the element a ∨ b has
rank 2. In particular, any two atoms in {a, b, c} form an empty 2-pyramid.
This is the reason why an atom of a graded relatively complemented lattice
may lie in 2γ empty 2-pyramids; see the bound in the following lemma. In
a polytope a vertex may lie on at most γ missing edges.
The bold part of Figure 8.6(b) also is an empty pyramid, as the only
element above the bold part has three additional atoms. (It is easy to
check that the lattice is graded and relatively complemented.) The reason
why these types of empty pyramids exist is that a statement similar to
Lemma 8.2.1 fails for graded relatively complemented lattices.
Lemma 8.10.2. Let L be a graded relatively complemented lattice of rank
r on r + γ atoms, a ∈ A(L), L′ := [a, 1̂], and γ ′ = γ(L′ ).
Let M = {M1 , . . . , Mn } be the set of empty pyramids Mi that have a
base xi ∈ Mi and an apex ai ∈ A(L) such that
• a ≤ xi , and
146
8.10. Empty Pyramids in Upper Intervals
• Mi′ := {y ∈ Mi : y ≥ a} is not an empty pyramid in L′ := [a, 1̂].
Then n ≤ 2(γ − γ ′ ).
Proof. The lattice L′ has r − 1 + γ ′ atoms. Each of these atoms is a rank
2 element in L that is above a and above at least one other atom in L.
Since L′ has r − 1 + γ ′ atoms, we can choose a set B ⊆ A(L) \ {a} of size
r − 1 + γ ′ such that every atom of L′ can be written as a ∨ b with b ∈ B.
Every atom in C := A(L) \ ({a} ∪ B) either (a) does not lie below a rank
2 element that covers a, or (b) lies below a rank 2 element that covers a
and that covers at least two other atoms. In the latter case, at most one of
the atoms lies in the set B. Therefore, if m denotes the number of atoms
in C that do not lie below a rank 2 element that covers a, we have at most
e + 2(γ − γ ′ − e) = 2(γ − γ ′ ) − e empty 2-pyramids in the set M.
Let I ⊆ {1, . . . , n} be the set of indices of empty pyrmaids in M that
are empty k-pyramids for some k ≥ 3. Then xi 6= a for every i ∈ I. Let
Ai := A(Mi ∨ a) \ (A(xi ) ∪ {a}). Since Mi is an empty pyramid, the atom
ai is not pyramidal over xi and we have Bi 6= ∅.
Since Mi is an empty pyramid and xi > a, the atom ai is pyramidal over
a, that is, A(a ∨ ai ) = {a, ai }. In particular, the element a′i := a ∨ ai is an
atom of L′ . The atom a′i is pyramidal over every element in [a, xi ] ⊆ L′ , as
Hi′ is not empty in L′ . Hence, for every b ∈ Bi the element b ∨ a is not in
A(L′ ) and we have |I| ≤ m.
We now show that whenever i 6= j, for i, j ∈ I, then Bi ∩ Bj = ∅. Let
bi ∈ Bi and bj ∈ Bj .
We have bi ∈ A(xi ∨ ai ), so a ∨ bi ≤ xi ∨ ai = xi ∨ a′i . Since bi 6≤ xi and
A(xi ∨ a′i ) = A(xi ) ∪ {a′i }, we have a′i ≤ a ∨ bi . The same holds for j, so we
have the following inequalities
a′i ≤ a ∨ bi ≤ xi ∨ a′i ,
a′j ≤ a ∨ bj ≤ xj ∨ a′j .
(8.1)
(8.2)
We distinguish two cases: a′i 6= a′j and a′i = a′j .
Case (i). If a′i 6= a′j , then [a′i , (xi ∧ xj ) ∨ a′i ] ∩ [a′j , (xi ∧ xj ) ∨ a′j ] = ∅, since
a′i is pyramidal over xi and a′j is pyramidal over xj . Clearly, this implies
that bi 6= bj .
Case (ii). If a′i = a′j , then ai = aj and both ai and aj are pyramidal over
a. Then obviously xi 6= xj .
Suppose a ∨ bi = a ∨ bj ∈ L′ . Then 8.1 and 8.2 imply that a ∨ bi ∈
[a′i , (xi ∧ xj ) ∨ a′i ], because a′i is pyramidal over xi and over xj .
But then there is w ∈ [a, xi ∧ xj ] such that w ∨ ai = a ∨ bi and thus ai is
not pyramidal over w < xi . This contradicts the fact that Mi is an empty
pyramid.
147
Thus, a ∨ bi 6= a ∨ bj and this implies bi 6= bj .
In total, we thus have that n ≤ 2(γ − γ ′ ) − m + m = 2(γ − γ ′ ).
8.11
Proof for Relatively Complemented Lattices
Theorem 8.11.1. Let L be a graded relatively complemented lattice of rank
r on r + γ atoms. Let k ≥ 1 and let Γ be the collection of all empty
ℓ-pyramids with ℓ ≤ k.
Then
|Γ| ≤ 2k γ k .
In particular, the number of empty ℓ-pyramids with ℓ ≤ k is bounded by a
function independent of r.
Proof. We prove the statement by induction on k. Every atom is pyramidal
over 0̂. Thus if k = 1 we have |Γ| = 0.
Let Br−γ ⊆ L be a boolean interval of rank r − γ, which exists by
Corollary 8.6.7. Let B = A(Br−γ ) and D = A(L) \ B. Then |B| = r − γ
and |D| = 2γ.
For every atom a ∈ A(L) the lattice La = [a, 1̂] is graded, and relatively
complemented. Let Γa be the collection of all empty ℓ-pyramids in La
with ℓ ≤ k − 1. By induction, |Γa | ≤ 2k−1 γ(La )k−1 . Additionally, by
Lemma 8.10.2, there are at most 2(γ − γ(La )) many empty pyramids that
contain a in one of their bases, but that do not appear as an empty pyramid
in La . Therefore, there are at most 2k−1 γ(La )k−1 + 2(γ − γ ′ ) ≤ 2k−1 γ k−1
empty ℓ-pyramids with ℓ ≤ k that have a base that contains a.
We claim that every empty ℓ-pyramid with ℓ ≤ k is counted at some
a ∈ D.
Let M be an empty ℓ-pyramid of L. Then there are b ∈ A(L) and
x ∈ L such that x is a base of M and b an apex. If A(x) ∩ D 6= ∅ and
a′ ∈ A(x) ∩ D, then M is counted at a′ .
Otherwise A(x) ⊆ B. Then [0̂, x] is boolean, and any other element
of M of rank ℓ − 1 is a base of M . Since M is an empty ℓ-pyramid with
ℓ ≤ r − γ and Br−γ is boolean of rank r − γ, there has to be z ∈ M of rank
ℓ − 1 such that A(z) ∩ D 6= 0,
Thus, the total number of empty ℓ-pyramids of L with ℓ ≤ k satisfies
|Γ| ≤ |D| 2k−1 γ k−1 = 2k γ k ,
as claimed.
Lemma 8.11.2. Let L be a pyramidally perfect lattice of rank r on r + γ
atoms. Every empty k-pyramid of L has at most k + γ − 1 atoms.
148
8.12. Proof for Pyramidally Perfect Lattices
Proof. The proof is the same as the one for polytopes; compare Lemma 6.2.2.
Indeed, an element x of rank (k − 1) has at most k + γ atoms, with equality
if and only if L arises from x by taking a pyramid r − k times.
Corollary 8.11.3 (Perles’ Skeleton Theorem for graded relatively
complemented lattices). For fixed k ≥ 1 and γ ≥ 0, the number of
combinatorial types of k-skeleta of graded relatively complemented lattices
of rank r on r + γ atoms is bounded.
Proof. Since, for fixed k and fixed γ, there is only a finite number of graded
relatively complemented lattices of rank at most r ≤ k + γ on r + γ atoms,
we can assume that k ≤ r − γ.
By Theorem 8.11.1, the number of empty ℓ-pyramids with ℓ ≤ k is
bounded by a function of k and γ. This implies that the size Kerk (L) is
bounded by a function of k and γ, as an empty k-pyramid has at most
k + γ − 1 atoms by Lemma 8.11.2.
Thus, also the number of combinatorial types of k-skeleta is bounded
by a function of k and γ, as the k-skeleton can by reconstructed, up
to isomorphism, from the k-kernel and the total number of atoms, by
Lemma 8.7.4.
8.12
Proof for Pyramidally Perfect Lattices
Definition 8.12.1 (Tetration). Let n ≥ 0. We define a function f (n) by
. . .2
f (n) := |22{z },
n
that is, recursively f (n) is defined by f (0) = 1 and f (n) = 2f (n−1) .
For k, γ ≥ 0 we write f (k, γ) = f (2k + γ − 1).
The following theorem can be seen as a generalization of Lemma 8.6.6,
and it follows the idea that the k-skeleton looks like the (direct) product
of a “small” part of the k-skeleton with a “large” boolean lattice; compare
Chapter 6.
Theorem 8.12.2. Let L be a pyramidally perfect lattice of rank r on r + γ
atoms and k an integer with 1 ≤ k ≤ r − γ.
Then there is a set of atoms Ak such that for every x ∈ skelk−1 (L) there
is y ∈ L with x ∗ Ak = y, and such that the size of A(L) \ Ak is bounded by
. . .2
f (k, γ) = |22{z } .
2k+γ−1
149
Proof. We construct the set Ak inductively. If k = 1, then according to
Lemma 8.6.6 there is y of rank r − γ and A1 = A(y) with 0̂ ∗ A1 = y. As
|A1 | ≥ r − γ, we have A(L) \ A0 ≤ 2γ ≤ f (1, γ).
Let k ≥ 1 and let Ak−1 be given with the desired properties.
Claim. Whenever some element x ∈ L of rank k does not have y ∈ L
with y = x ∗ (Ak−1 \ A(x)), then this x is a join of atoms in W :=
A(L) \ Ak−1 .
Proof of the claim. Let x ∈ L be of rank k and set W ′ := W ∩ A(x). Let
_
w′ .
z :=
′
′
w ∈W
If rk(z) = k, then x = z and x is a join of elements in W . Otherwise,
we have rk(z) < k and by the definition of Ak−1 there is y with
y = z ∗ (Ak−1 \ A(z)). Clearly, x = z ∗ A′ for A′ = A(x) ∩ Ak−1 and
consequently y = x ∗ (Ak−1 \ A(x)).
Let x1 , . . . , xn be the elements of rank k such that no y ∈ L exists with
y = xi ∗ (Ak−1 \ A(xi )). Every such element is the join of atoms in the
set W , and the size of W is bounded by f (k − 1, γ). Since the lattice L is
atomic, the number n is bounded by 2f (k−1,γ) .
By Lemma 8.6.6, there are elements y1 , . . . , yn of rank r − γ and sets of
atoms B1 , . . . , Bn ⊆ A(L) with
y1 = x1 ∗ B1 , y2 = x2 ∗ B2 , . . . , yn = xn ∗ Bn .
Let B = B1 ∩ B2 ∩ · · · ∩ Bn . The size of every Bi is bounded from below
by |Bi | ≥ (r − γ) − (k + γ) = r − (k + 2γ), since an element of rank k
has at most k + γ atoms below it. Hence, for every Bi , i = 1, . . . , n, in the
intersection we “lose” at most (k + 3γ) of the atoms of L. Then the size of
B is bounded from below by
|B| ≥ r + γ − n(k + 3γ).
We set Ak := Ak−1 ∩ B. Then for every element x of rank k there is
y ∈ L with y = x ∗ Ak . The size of Ak is bounded from below by
|Ak | ≥ |Ak−1 | − n(k + 3γ).
Thus, the size of A(L) \ Ak is bounded from above by
r + γ − |Ak | ≤ r + γ − |Ak−1 | + n(k + 3γ)
≤ f (k − 1, γ) + 2f (k−1,γ) (k + 3γ)
≤ f (k, γ),
as claimed.
150
8.12. Proof for Pyramidally Perfect Lattices
Corollary 8.12.3 (Perles’ Skeleton Theorem for pyramidally perfect lattices). For fixed k ≥ 1 and γ ≥ 0, the number of combinatorial
types of k-skeleta of pyramidally perfect lattices of rank r on r + γ atoms is
bounded.
Proof. Since, for fixed k and fixed γ, there is only a finite number of pyramidally perfect lattices of rank at most r ≤ k + γ on r + γ atoms, we can
assume that k ≤ r − γ.
By Theorem 8.12.2, there is a set of atoms A such that for every x ∈
skelk (L) there is y ∈ L with x ∗ A = y, and such that the size of A(L) \ A
is bounded by f (k, γ). Clearly no element of A lies in the k-kernel and so
the size of Kerk (L) is as well bounded by f (k, γ).
Thus, also the number of combinatorial types of k-skeleta is bounded
by a function of k and γ, as the k-skeleton can by reconstructed, up
to isomorphism, from the k-kernel and the total number of atoms, by
Lemma 8.7.4.
151
Part III
Chapter 9
Nonsimplicial Mani Polytopes
The theory of illumination of convex bodies, which has its roots in papers
by Soltan [104] and Grünbaum [49], is the main source for the material in
this chapter. In particular, the result presented here relates to a problem
on illuminated polytopes (that is, polytopes in which every vertex lies on an
inner diagonal) that was posed by Hadwiger in 1972 [52]. He describes the
problem as follows:
Vermutlich gilt die folgende Aussage: Hat ein Polytop, also ein kompaktes
konvexes Polyeder P , des n-dimensionalen euklidischen Raumes die Eigenschaft, dass sich zu jeder seiner Seitenflächen noch wenigstens eine andere
mit ihr disjunkte Seitenfläche aufweisen lässt, so gilt für die Anzahl f der
Seitenflächen von P die Ungleichung f ≥ 2n. Offensichtlich gilt Gleichheit beim Hyperwürfel, also beim 2n-Zell. Demnach ist also 2n die kleinste
mögliche Seitenflächenzahl für Polytope der oben genannten Eigenschaft.1
He goes on with an argument that his conjecture is true for dimensions
n ≤ 3. Only the four combinatorial types that are displayed in Figure 9.1
are candidates for counterexamples—none of them is one, as none of them
satisfies the “disjoint facets condition.”
He then acknowledges the difficulty of the general problem:
1
The following statement is probably true: If a polytope, that is, a compact convex
polyhedron P , in n-dimensional Euclidean space has the property that for every facet one
can find a disjoint facet, then the number f of facets of P satisfies f ≥ 2n. Obviously, we
have equality for the hypercube, that is, for the 2n-cell. Accordingly, 2n is the smallest
possible number of facets of polytopes with this property.
155
9. Nonsimplicial Mani Polytopes
(a) A 2-simplex.
(b) A 3-simplex.
(c) A pyramid over a quadrilateral.
(d) A prism over a triangle.
Figure 9.1: A complete list of polytopes in dimensions n = 2 and n = 3 with
less than 2n facets.
Seltsamerweise scheint es, dass die Abklärung, ob unsere Vermutung für
alle Dimensionen n richtig ist oder nicht, viel schwieriger ist, als ein Konvexgeometer bei erster Konfrontation mit der Frage anzunehmen geneigt
ist. Bereits einige haben sich vergeblich bemüht.2
He concludes by mentioning a letter by Grünbaum, in which the latter
confirms the conjecure for n = 4, and by asking, whether the conjecture is
true for all dimensions.
(We now switch back from n to d to denote dimension.)
2
Strangely, it seems that the clarification, whether our conjecture is true for all
dimensions n or not, is much more difficult than a convex-geometer would be inclined to
believe upon first confrontation with this problem. Already several have tried in vain.
156
9.1. Illuminated Polytopes
Hadwiger’s question asks, in its polar-dual formulation, whether an illuminated d-polytope has at least 2d vertices. The obvious examples with
equality are the d-crosspolytopes.
Mani settled this question in an extraordinary paper from 1974 [70].
Not only did he give a negative answer to Hadwiger’s question, by constructing simplicial illuminated d-polytopes on fewer than 2d vertices. He
also determined the extremal function s(d) for the number of vertices of
such √
a polytope precisely. As it turns out, this function is roughly given by
d + 2 d, that is, for large d there are illuminated polytopes on much fewer
vertices than anticipated; see Theorem 9.1.4 below.
Mani’s construction of illuminated d-polytopes on s(d) vertices is easy
to understand—we will review it below. Proving the tight lower bound
for the function s(d) is much harder, even with a simplification found by
Rosenfeld [96]. The argument consists to a large part of a series of delicate
geometric “repositionings”: Given an illuminated polytope on the minimum
number of vertices, one can carefully move the vertices to obtain a simplicial
illuminated polytope of specific combinatorial type that is similar to those
constructed by Mani. For these it is easy to show that they have at least
s(d) vertices.
In this chapter we are concerned with the set of illuminated polytopes
on s(d) vertices and a question by Mani related to this set. His extremal examples constructed in [70] are simplicial and he asked whether nonsimplicial
such polytopes exist.
We give a precise answer to Mani’s question. While for d ≤ 5 the only
extremal illuminated polytopes are the d-crosspolytopes, we show that for
every d ≥ 6 there exists a nonsimplicial one.
9.1
Illuminated Polytopes
In this section we define illuminated polytopes and state Mani’s result on
the extremal function of the number of vertices of these polytopes [70]. All
polytopes in this chapter are assumed to be full-dimensional.
Suppose that we have a polytope, and imagine that this polytope is hollow. Think of each vertex of the polytope as an infinitesimally small light
source that casts light on those points of the boundary that one can “see”
from the vertex. Thus, a vertex does not cast light on all the points of
proper faces of the polytope that contain that vertex. In particular, it does
not cast light on itself. If all vertices are illuminated by the light cast from
other vertices, then also the whole boundary is illuminated. Indeed, convex combinations of illuminated points are again illuminated. This image
157
explains the choice of the word “illuminated” in the following definition.
Definition 9.1.1 (Inner diagonal, illuminated). Let P be a d-polytope
in Rd . For u, v ∈ Rd , the line segment
[u, v] := {λu + (1 − λ)v : 0 ≤ λ ≤ 1}
is an inner diagonal of P if u and v are vertices of P and [u, v] ∩ int P 6= ∅,
that is, the segment [u, v] hits the interior of P .
The polytope P is called illuminated if every vertex of P lies on an inner
diagonal.
A more general definition of diagonals that comprises the definition of
inner diagonals can be given; see Section 10.2.3.
Inner diagonals of polytopes were studied extensively by Bremner &
Klee [31]. They obtained two upper bound theorems on the number of
inner diagonals of general polytopes.
They showed that, for fixed d and f0 , the maximum number of inner
diagonals is
µ ¶
µ
¶
f0
d+1
− df0 +
,
2
2
and that this is only attained by stacked polytopes [31, Theorem 3.9]. This
is the same number as the maximum number of missing edges of a simplicial
polytope, by the Lower Bound Theorem [7] [9]. In the proof of the upper
bound for inner diagonals we can assume that the polytope is simplicial, as
a perturbation of the vertices only increases the number of inner diagonals.
This is similar to the first step in the proof of the Upper Bound Theorem
by McMullen [76].
Bremner & Klee also showed that, for fixed d and fd−1 , the maximum
number of inner diagonals is attained by certain simple polytopes [31, Theorem 3.10]. In addition, they analyzed with great care inner diagonals of
3-polytopes.
We are interested in the minimal number of vertices of an illuminated
polytope and the set of polytopes that attain this number.
Definition 9.1.2 (Mani polytopes). For d ≥ 1 we define the parameter
s(d) := min{f0 (P ) : P is an illuminated d-polytope},
that is, s(d) denotes the minimum number of vertices of an illuminated dpolytope. We call an illuminated polytope on s(d) vertices a Mani polytope.
158
9.1. Illuminated Polytopes
Definition 9.1.3. For d ≥ 1, we set p(d) := ⌈
√
4d+1−1
⌉.
2
The term p(d) shows up in the function of the minimum number of
vertices of illuminated polytopes. The following theorem is the main result
in Mani’s paper [70].
Theorem 9.1.4 (Mani [70]). For every d ≥ 1,
s(d) = min{2d, d + p(d) + ⌈d/p(d)⌉ + 1}.
For d ≤ 5, the function s(d) equals 2d, for d = 6, 7 it equals 2d =
d + p(d) + ⌈d/p(d)⌉ + 1, and for d ≥ 8 it equals d + p(d) + ⌈d/p(d)⌉ + 1 < 2d.
The value of p(d) is one possible solution to finding a positive integer p
such that p + ⌈d/p⌉ is minimized for fixed d. The global minimum
√ of the
function f (x) = x + d/x in the continous variable x is given by d. The
integer solution is not unique in general: If d = 6, then both p = 2 and
p = 3 attain the minimum of 5.
A given p is a minimizer for the term p + ⌈d/p⌉ at least for every d
in the range p(p − 1) < d ≤ p(p + 1), that is, p + ⌈d/p⌉ ≤ q + ⌈d/q⌉ for
every positive integer q. The following lemma then implies that p(d) is a
minimizer for p + ⌈d/p⌉.
Lemma 9.1.5. Let p := p(d) = ⌈
hold:
√
4d+1−1
⌉.
2
Then the following inequalities
p(p − 1) < d ≤ p(p + 1).
Proof. We prove the upper bound first, as the calculation is shorter. We
use that ⌈x⌉ + k = ⌈x + k⌉ for all x ∈ R and all integers k.
»√
¼ »√
¼
4d + 1 − 1
4d + 1 + 1
p(p + 1) =
2
2
¶ µ√
¶
µ√
4d + 1 − 1
4d + 1 + 1
≥
2
2
=d
For the strict lower bound, we first calculate a (nonstrict) lower bound, and
argue later why this implies a strict lower bound. We use that ⌈x⌉ ≤ ⌊x+1⌋
159
for all x ∈ R.
»√
¼ »√
¼
4d + 1 − 1
4d + 1 − 3
p(p − 1) =
2
2
º ¹√
º
¹√
4d + 1 + 1
4d + 1 − 1
≤
2
2
¶ µ√
¶
µ√
4d + 1 + 1
4d + 1 − 1
≤
2
2
=d
(9.1)
(9.2)
√
Note that the bound in (9.1) is strict if and only if 4d+1+1
is an integer,
2
which is the only case in which (9.2) is not strict. That is, there is at most
equality in (9.1) or in (9.2), but not both.
Why does the minimum of p + ⌈d/p⌉ show up in the function s(d)? This
is partially answered by the construction of the extremal polytopes, and we
get back to this question after the discussion of the construction.
Before we move on, I want to mention a related open problem on illuminated polytopes.
Definition 9.1.6 (Primitively illuminated). Let P be a d-polytope.
Then P is said to be primitively illuminated by its vertices if P is illuminated
and no proper subset of V(P ) illuminates P . That is, there is no proper
subset U ⊂ V(P ) such that for every v ∈ V(P ) there is u ∈ U such that
[u, v] is an inner diagonal of P .
One can show that a d-polytope is primitively illuminated by its vertices
if and only if the graph of inner diagonals, that is, the graph
Ginn (P ) := (V, {(u, v) : [u, v] is an inner diagonal of P })
is a perfect matching on V .
Question 9.1.7 (Boltiyanski, Martini & Soltan [29]). Is there a bound
on the number of vertices of a primitively illuminated d-polytope? (The dcube is primitively illuminated by its vertices. However, no example on more
than 2d vertices is known.)
9.2
Mani’s Simplicial Illuminated Polytopes
For reference and comparison we review Mani’s construction of simplicial
Mani polytopes [70].
160
9.2. Mani’s Simplicial Illuminated Polytopes
Let Cd (n) := conv({(t, t2 , . . . , td ) : t ∈ [n]}) be the cyclic d-polytope on
n vertices {v1 , . . . , vn }.
We need Gale’s evenness criterion, which gives a combinatorial description of the cyclic polytopes.
Theorem 9.2.1 ([118, Theorem 0.7]). The cyclic d-polytope Cd (n) on
n vertices has the following properties:
(i) It is a simplicial polytope.
(ii) (Gale’s evenness criterion) A set S ⊆ [n] with |S| = d is the index set
of the vertices of a facet of Cd (n) if and only if
for all i < j, i, j ∈
/ S : |{k ∈ S : i < k < j}| is even.
The basic idea of Mani’s construction is the following: Suppose that Q
is a d-polytope, and F a set of facets of Q such that the complements of
the facets in F cover all vertices. Then stacking a vertex onto every facet
in F, we obtain an illuminated polytope. The number of vertices of this
polytope is f0 (Q) + | F |. We want this number to be small, so Q should
not have too many vertices and the facets in F should ideally be disjoint.
Let d ≥ 2, p ≥ 1, and set q := ⌈d/p⌉. We construct an illuminated
polytope by stacking q + 1 times onto a cyclic d-polytope on d + p vertices,
Let Q := Cd (d + p) and let {v1 , . . . , vd+p } be the vertex set of Q. For
every j ∈ {1, . . . , q} we set
Cj := {n ∈ N : (j − 1)p + 1 ≤ n ≤ jp},
and
Cq+1 := {n ∈ N : d + 1 ≤ n ≤ d + p}.
Then the following hold:
(i) For every j ∈ {1, . . . , q + 1} the set [d + p] \ Cj is the index set of a
facet, by Gale’s evenness criterion; see Theorem
S 9.2.1.
(ii) The sets Cj cover the vertices of Q, that is, qj=1 Cj = V(Q).
Denote by Fj the facet with vertex set V(Q)\{vi : i ∈ Cj }, and let I d (P )
be the polytope obtained from Q by stacking a vertex onto each of the
facets F1 , . . . , Fq+1 . The polytope I d (p) is illuminated “by construction,”
as a stacked vertex lies on inner diagonals with all vertices that lie in the
complement of the stacked facet. If we set p := p(d), the polytope I d :=
I d (p) is a Mani polytope by Theorem 9.1.4 for d ≥ 5. Since Cd (p) is
simplicial, so is I d .
161
(b) The polytope I 3 .
(a) The polytope I 2 .
Figure 9.2: The polytopes that result from Mani’s construction in dimensions
2 and 3.
Example 9.2.2. Let us look at examples in low dimensions.
For d = 2, 3, the polytope I d is obtained from a simplex by stacking all
facets. Thus, the number of vertices is 2(d + 1) and “we get nothing but
pretty pictures”; see Figure 9.2.
For d = 4, 5, the polytope I d has one vertex more than the d-dimensional
crosspolytope.
For d = 6, 7, the polytope I d is an illuminated polytope on the same
number of vertices as the d-crosspolytope, which is 2d. It is not isomorphic
to the crosspolytope, though: The vertices of the crosspolytope all have the
same degree of 2d−1 > d and a stacked vertex of I d only has degree d.
For d ≥ 8, Mani’s polytopes have strictly fewer vertices than the crosspolytopes, so d = 8 is the first interesting case. We have p(8) = 3, so we
need to stack onto some of the facets of C8 (11). The sets Cj , j = 1, . . . , 4,
are schematically displayed in the following figure:
1| {z
2 3}
C1
4| {z
5 6}
C2
z
7| {z
8 9}
C3
C
}|4 {
10 11.
The four sets C1 , C2 , C3 , C4 are facet complements of C8 (11). As their
union covers all vertices of C8 (11), the resulting polytope after stacking is
illuminated. It has 11 + 4 = 15 < 16 = 2d vertices.
With the above construction, we have proven the following theorem by
Mani [70].
Theorem 9.2.3 (Mani [70]). Let d ≥ 2. Then there is a simplicial illuminated d-polytope I d on d + p(d) + ⌈d/p(d)⌉ + 1 vertices.
162
9.3. Nonsimplicial Mani Polytopes
As we have shown in Section 9.1, we cannot improve the construction
by choosing a different p. Of course, we know by Theorem 9.1.4 that the
polytopes I d have the minimum number of vertices for d ≥ 5. A first step
towards a proof of that theorem is to show that the polytopes I d have the
minimum number of vertices among all illuminated polytopes constructed
in a similar way [70, Lemma 1]. We do not go further in that direction, as
we have other plans.
9.3
Mani’s extremal illuminated polytopes are simplicial, and he asked whether
nonsimplicial ones exist.
Question 9.3.1 (Mani [70, p. 66]). Are there nonsimplicial Mani polytopes?
This question is also mentioned by Bremner & Klee [31], with an erroneous answer for low dimensions. They claim that it follows from Mani’s results in [70] that for d ≤ 7 the only Mani polytopes are the d-crosspolytopes.
It is true in dimensions d ≤ 7 that the set of Mani polytopes includes the
d-crosspolytope. However, as we have seen in Example 9.2.2 there are at
least two combinatorial types of simplicial Mani polytopes in dimensions
d = 6, 7. We show below that there are even nonsimplicial ones.
9.3.1
Unique Mani Polytopes
For dimensions d = 3 and d = 4 one can make complete lists of all polytopes
on at most 2d vertices, which is the interesting range for the number of
vertices in the case of illuminated polytopes.
For d = 3 making such a list is an exercise for an introductory class on
polytopes [119].
For d = 4 one can do the enumeration up to 7 vertices using Gale
diagrams by hand [51, pp. 112–113]. The 4-polytopes on 8 vertices can
be enumerated [2], but already because of their large number (according
to Altshuler & Steinberg [2] there are 1294 different combinatorial types)
one certainly needs the help of a computer to accomplish this task; see
also [103].
Checking these lists, one finds that the only illuminated polytopes in
dimensions d = 3, 4 on 2d vertices are the d-crosspolytopes.
We show here that in dimensions 1 ≤ d ≤ 5 Mani polytopes are combinatorially unique. Thus the only type that appears is the d-crosspolytope.
This can be inferred from results by Mani [70] and Rosenfeld [96].
163
Definition 9.3.2 (Self illuminated, opposite sets). Let P be a dpolytope. A set of vertices U ⊆ V(P ) is said to illuminate itself if for
every vertex v ∈ U there is a vertex u ∈ U such that [u, v] is an inner
diagonal.
A set W ⊆ V(P ) is said to lie opposite the vertex v ∈ V(P ) if for
every w ∈ W the segment [v, w] is an inner diagonal and V(P ) \ (W ∪ {v})
illuminates itself.
Let Γ(P ) := max{|W | : W lies opposite some v ∈ V(P )}.
The following is the main result from [96] with a slightly stronger statement that is easily extracted from Rosenfeld’s proof.
Theorem 9.3.3 (Rosenfeld [96]). Let P be an illuminated d-polytope.
If Γ(P ) = 1, then f0 ≥ 2d and there is a perfect matching on the inner
diagonals.
The next lemma is easily derived from results by Mani [70, Lemma 1,
Proposition 2, and Proposition 3].
Lemma 9.3.4 (Mani [70]). Let d ≥ 3, let P be a Mani d-polytope, and
assume that Γ(P ) ≥ 2. Then f0 (P ) ≥ d + p(d) + ⌈d/p(d)⌉ + 1.
Corollary 9.3.5. Let d ≥ 3, let P be a Mani d-polytope with Γ(P ) ≥ 2.
Then d ≥ 6.
Proof. By Lemma 9.3.4, the number of vertices is at least d + p(d) +
⌈d/p(d)⌉ + 1. But for 3 ≤ d ≤ 5 we have that d + p(d) + ⌈d/p(d)⌉ + 1 > 2d.
Since P is a Mani polytope we must have d ≥ 6.
Theorem 9.3.6. There is exactly one combinatorial type among all Mani
d-polytopes for 1 ≤ d ≤ 5. It is given by the d-crosspolytope.
Proof. The cases d = 1, 2 are trivial, so assume d ≥ 3. Let P be a Mani
polytope with 3 ≤ d ≤ 5.
By Corollary 9.3.5, we have Γ(P ) = 1. Theorem 9.3.3 and existence of
the crosspolytopes then imply that f0 (P ) = 2d and that there is a perfect
matching on the inner diagonals. Since any facet of P can contain only one
vertex of any inner diagonal, the set of facets is a subset of the facets of the
d-crosspolytope. This implies that P is the d-crosspolytope.
9.3.2
Nonsimplicial Mani polytopes can be constructed in different ways.
164
One method that works for some dimensions is the pseudo-stacking operation, described, for example, by Paffenholz & Werner [86]; see also [1].
In all dimensions except for d = p2 and d = p(p + 1), Mani’s construction
leaves some room for local modifications. In these cases one can move one
of the stacked vertices into a facet defining hyperplane, and this results in
a nonsimplicial Mani polytope.
We describe in this section a unified construction that settles the question for all dimensions. Recall that we want to cover the vertices by facet
complements of maximal size, that is, we want to construct a d-polytope on
d + p vertices that has “many” disjoint facet complements of size p. If we
consider Gale diagrams, this problem becomes nearly trivial, as the positive
circuits of a Gale diagram correspond exactly to the facet complements of
the polytope. The construction in the proof of the following theorem follows
this idea.
Theorem 9.3.7. There exists a nonsimplicial Mani d-polytope in every
dimension d ≥ 6.
Proof. For every d ≥ 6 we construct a nonsimplicial Mani d-polytope (recall
that for d = 6, 7 we have d+p(d)+⌈d/p(d)⌉+1 = 2d). Let p ≥ 1, q := ⌈d/p⌉,
and choose a k with 1 ≤ k ≤ q − 1.
We construct a nonsimplicial polytope Q that has q + 1 simplex facets,
such that stacking onto these facets produces a nonsimplicial illuminated dpolytope. (What we describe here is in fact a whole family of such polytopes,
indexed by the parameter k.)
We describe Q in terms of a Gale diagram A. Let
B = {e1 , . . . , ep−1 , − 1},
where 1 denotes the vector in which all entries are 1. This is a positive
basis of Rp−1 of cardinality p. The vectors in A are the following:
(1) Take k copies of B, and denote them by B1 , . . . , Bk .
(2) Take q − k copies of −B, and denote them by B̃1 , . . . , B̃q−k .
(3) Furthermore, take the vectors − 1, e1 , . . . , ed+p−pq−1 one more time.
Then the number of vectors in A is d + p.
By Theorem 1.3.3, every Bi , i = 1, . . . k, and every B̃j , j = 1, . . . , q − k
corresponds to facet complements of size p in Q, that is, to complements of
simplex facets . If we augment the set {− 1, e1 , . . . , ed+p−pq−1 } to a positive
basis B ′ by taking the last pq − d vectors of B1 we have that
{Bi : i = 1, . . . , k} ∪ {B̃j : j = 1, . . . , q − k} ∪ {B ′ }
165
is a set of subconfigurations of A that correspond to complements of simplex
facets of Q. These complements cover all vertices of Q. Stacking onto the
corresponding facets we obtain an illuminated polytope P on d + p + q + 1
vertices. For p = p(d) we get that P is a Mani polytope.
If 1 ≤ k ≤ q − 1, then there is a set of two vectors in A that correponds
to a facet complement, so Q is nonsimplicial, unless p = 2. For d ≥ 7, we
have p(d) ≥ 3 and we indeed get a nonsimplicial polytope. However, for
d = 6 we get p = p(d) = 2 and q = 3. In this case, we choose p = 3 instead
of p(d). Then q = ⌈d/p⌉ = 2 and we have f0 (P ) = 12 = s(6), that is, P is
a Mani polytope.
In both cases, the polytope P is nonsimplicial, because Q is nonsimplicial and we only stack onto simplex facets.
Example 9.3.8. We look at two examples that arise from the above description.
For d = 6, we get a polytope Q as in the proof of Theorem 9.3.7 by
constructing the Gale diagram in Figure 9.3(a) with p = 3, q = 2, and
k = 2.
We have seen this polytope before: It is the example of a 6-polytope
Lockeberg constructed in [69] to show that in general simplex refinements
cannot have two prescribed principal vertices; see Figure 2.1.
The Gale diagram has three disjoint positive bases that cover all vectors:
the bases B1 = B2 = {e1 , e2 , − 1} and the basis B̃1 = {−e1 , −e2 , 1}. These
bases correspond to complements of simplex facets of Q. Stacking onto these
three facets produces a nonsimplicial illuminated 6-polytope on s(6) = 12
vertices.
For d = 16, we display the result of the construction as an affine Gale
diagram in Figure 9.3(b), where all points that “touch” represent different
copies of a single point. (See [118, Chapter 6] for affine Gale diagrams and
how they are related to ordinary Gale diagrams.)
In this case, the polytope Q has f0 = 20 and five disjoint simplex facet
complements of size four that cover all vertices. This yields a nonsimplicial
illuminated 16-polytope on s(16) = 25 vertices.
The following remarks are due to McMullen [79].
The construction is highly modifiable: Every vector configuration that
contains at least two disjoint positive bases is a Gale diagram of a polytope.
Thus, one may arbitrarily put the right number of positive bases of size p
in Rp−1 (and double some vectors to get the right number of vectors).
The polytopes constructed by Gale diagrams in the proof of Theorem 9.3.7 have a simple direct description. They can be obtained by applying a series of vertex splits on a prism over a simplex of suitable dimension.
166
2
2
2
(b) Stacking onto the right set of facets of
a polytope with this Gale diagram yields
a nonsimplicial Mani 16-polytope.
(a) Stacking onto the right set of facets a
polytope with this Gale diagram yields a
nonsimplicial Mani 6-polytope.
Figure 9.3: Gale diagrams of building blocks for nonsimplicial Mani polytopes.
Consider the following

1
1

2
1



..

.


 0

p+1  1 1

p+2  1

..
1

.


2p
matrix of size 2p × 2p:
0
0
..
.
...
1
1
1
...
0
1
···
0
1
···
··· 1 1
1
1 ··· 1
−1 −1
−1
−1
..
...
...
.
1 −1
−1
0
0
..
.
0
1
1
1
1
..
.
1

















The first p + 1 rows contain as columns the vertices of the standard prism
over a (p − 1)-simplex lifted to the affine hyperplane with xp+1 = 1 in Rp+1 .
The last p − 1 rows contain as columns the vectors of a Gale diagram
of this polytope. It consists of a positive basis in Rp−1 of size p and its
negative. Doubling a vector of this Gale diagram corresponds to a vertex
split at the corresponding vertex.
167
Chapter 10
Counterexamples to
Marcus’ Conjecture
A minimal positive k-spanning vector configuration is a positively spanning
vector configuration that is still positively spanning after the deletion of
any (k − 1) arbitrarily chosen vectors and is inclusion-minimal with respect
to this property.
Marcus conjectured in [71] that the size of a minimal positive k-spanning
vector configuration in Rr is bounded by 2kr.
Why is this conjecture plausible?
For k = 1 it is the classical Blumenthal–Robinson Theorem [26]; see also
Theorem 10.2.1 below. Shephard gave an elegant Gale duality proof of this
theorem [101].
The corresponding statement for linear k-spanning configurations, which
was proven by Marcus [71], implies Marcus’ conjecture for centrally symmetric vector configurations.
Furthermore, Dalmazzo [36] has shown a related result in graph theory
on minimally k-edge-connected multidigraphs. A simpler proof was given
by Berg & Jordán; see [14, Theorem 3]. Dalmazzo’s result implies Marcus’
conjecture for vector configurations that can be interpreted as the incidence
matrix of such a multidigraph; we refer the reader to Marcus [71] for details.
For k = 2, Marcus’ conjecture, in its Gale-dual formulation for polytopes, directly relates to Hadwiger’s problem on illuminated polytopes: A
minimal positive 2-spanning configuration is a Gale diagram of an unneighborly polytope, a polytope in which every vertex lies on a missing edge. Thus
169
10. Counterexamples to Marcus’ Conjecture
in particular, Gale diagrams of illuminated polytopes are minimal positive
2-spanning configurations. The conjectured bound of 4m on the size of a
minimal positive 2-spanning configuration translates to a lower bound of
⌈4(d + 1)/3⌉ on the number of vertices of an unneighborly polytope.
In a subsequent paper by Marcus [73], the following short note appeared
at the end:
Note added in proof
As this goes to press, the author has discovered an unneighborly polytope
of dimension 36 having only 49 vertices.
(For d = 36, the conjectured bound of ⌈4(d + 1)/3⌉ equals 50.)
Unfortunately, Marcus did not give any hint how he did obtain such
a polytope. I tried to contact Marcus by writing to the California State
Polytechnic University in Pomona in 2007. He had been on the faculty
there for a number of years. Sadly, I received the information that he had
died some years ago.
Marcus’ counterexample is also mentioned in the “Mathematical Reviews” and “Zentralblatt” reviews of Marcus paper [73] by McMullen. However, McMullen told me that he had never seen the counterexample himself [79]. So it seems that Marcus’ counterexample is lost.
However, as noted, illuminated
√ polytopes are unneighborly, and as such
polytopes exist on roughly d+2 d vertices, it is clear that for large enough d
Marcus’ conjecture fails for k = 2. And indeed, Mani’s illuminated polytope
in dimension 36 has exactly 49 vertices. So, it is likely that Marcus’ example
was similar.
In this chapter we work out the connection between Mani’s illuminated
polytopes and minimal positive k-spanning configurations and prove the
following two main results:
• Starting from Mani’s simplicial illuminated polytopes on s(d) vertices,
or from our nonsimplicial ones constructed in Chapter 9, we give
counterexamples to Marcus’ conjecture for all k ≥ 2. Although nearly
trivial for even k, the construction requires some care for odd k.
• We prove an upper bound on the size of minimal positive k-spanning
configurations in Rr , adding to a result by Marcus on the size of
minimal positive 2-spanning configurations [71]. Mani’s illuminated
polytopes imply that there is no bound on the size of minimal positive
k-spanning configurations that is linear in r if k ≥ 2. Our bound is,
for fixed k, polynomial in r, and we achieve this with an application
of Perles’ Skeleton Theorem.
170
10.1. Unneighborly Polytopes
We also look at classes of unneighborly polytopes for which the conjectured
bound holds; see Section 10.2.3.
10.1
Davis’ paper [37] is a seminal piece for the theory of positive linear dependence with important classification results. As he remarks, the theory of
pointed cones reduces to the study of polytopes, whereas we know now,
with the advent of diagram techniques, that also the theory of solid cones,
as he calls them, is essentially the theory of polytopes.
In this section we work out the connection between minimal positive kspanning configurations and a certain class of polytopes using Gale duality.
Let V = (v1 , . . . , vn ) be a vector configuration in Rr . Recall from Section 1.3.5 that V is said to be positively spanning for a vector space W ⊆ Rr
if nonnegative combinations of vectors of V span the space W .
We call V a positive spanning configuration if it is positively spanning
for Rr .
The configuration V is a minimal positive spanning configuration, also
called a positive basis, if V is inclusion-minimal with respect to being positively spanning, that is, every proper subconfiguration of V is not positively
spanning for Rr (it might be positively spanning for its linear span).
Marcus [71] [73] generalized the notion of positively spanning in the
same way as k-connectedness of graphs generalizes connectedness.
Definition 10.1.1 (Positively k-spanning). For k ≥ 1, we call a vector
configuration V positively k-spanning if for every U ⊆ V of size at most
k − 1 the configuration V \ U is positively spanning.
We call V a minimal positive k-spanning configuration if V is inclusionminimal with respect to being positively k-spanning.
The notion of positively k-spanning can be interpreted in terms of polytopes. To do so, we need the following definition of k-unneighborly polytopes.
Definition 10.1.2 (k-unneighborly). Let P be a d-polytope with the
following two properties:
(i) It is (k − 1)-neighborly, that is, every set of k − 1 vertices is the set of
vertices of a simplex (k − 2)-face.
(ii) Every vertex is a vertex of an empty (k − 1)-simplex.
Then P is called k-unneighborly. If k = 2, we call P unneighborly.
171
If V is a vector configuration in Rr that is positively k-spanning for
k ≥ 2, we can associate to V a polytope P of dimension n − r − 1 on n
vertices that has V as a Gale diagram. This polytope is (k − 1)-neighborly,
according to the criterion in Theorem 1.3.3: Whenever we remove at most
k − 1 vectors from V , the remaining vectors are still positively spanning.
If V is a minimal positive k-spanning vector configuration, then P is
k-unneighborly: For every vector v ∈ V there are k − 1 vectors U , such
that the vectors in V \ (U ∪ {v}) are not positively spanning for their linear
span, as a minimal positive spanning configuration minus one vector is still
linearly spanning for the whole space. By Theorem 1.3.3, the vertices that
correspond to U ∪ {v} do not form a face of P .
10.2
The Sizes of Minimal Positive Spanning Configurations
We are interested in the size of minimal positive k-spanning configurations.
Throughout the rest of the chapter, we denote the size of the minimal
positive k-spanning configuration V by n.
A lower bound for n is trivially given by 2k + r − 1: Take a hyperplane
spanned by r − 1 vectors of the configuration. Then the positive half-space
and the negative half-space bounded by this hyperplane contain each at
least k of the vectors of V , as V is positively k-spanning. It was shown by
Marcus [73] that this lower bound is attainable. This result is better known
as Gale’s Lemma [43]; see for example [75, pp. 64–66].
Harder to get are upper bounds on the size of V . We begin with a
classical result for k = 1.
10.2.1
The Blumenthal-Robinson Theorem
For k = 1 we have the following classical result, which was first shown by
Blumenthal & Robinson [26], according to Davis [37, p. 744]. We therefore
call it the Blumenthal–Robinson Theorem.
Theorem 10.2.1 (Blumenthal & Robinson [26], Davis [37], Shephard [101]). If V is a positive basis for Rr of size n, then r + 1 ≤ n ≤
2r. If n = 2r, then the configuration is given by a basis (b1 , . . . , br ) and
(λ1 b1 , . . . , λr br ), where λi < 0 for i = 1, . . . , r.
Shephard’s proof [101] of this theorem is the simplest and most conceptual: He considers V as a Gale transform of a full-dimensional affine point
set P in (n−r−1)-dimensional space. Since for any v ∈ V the configuration
172
10.2. The Sizes of Minimal Positive Spanning Configurations
V \ {v} is not positively spanning, all points of P are double-points (one
might consider these as empty vertices). Thus we have that n ≥ 2(n − r),
that is, n ≤ 2r.
If equality holds in this equation, then P is an (n − r − 1)-simplex with
doubled vertices and any Gale transform of P has the desired form.
Thus, the Blumenthal–Robinson Theorem translates to the following
nearly trivial statement about affine point sets: If every point of a ddimensional affine point configuration lies on an empty vertex, that is, a
double point of this configuration, then this point configuration has at least
2d + 2 vertices.
10.2.2
Marcus’ Conjecture and Unneighborly Polytopes
Marcus conjectured [71] that the result by Blumenthal & Robinson [26]
generalizes to minimal positive k-spanning vector configurations in the following way.
Conjecture 10.2.2 (Marcus [71] [73]; disproved (Corollary 10.3.2)).
Let V ⊆ Rr be a minimal positive k-spanning vector configuration of size
n. Then n ≤ 2kr. If n = 2kr, then V is contained in r lines through the
origin.
By taking k copies of a linear basis and k copies of its negative, we
obtain a minimal positive k-spanning configuration on exactly 2kr vectors.
Marcus [71] proved his conjecture for r = 2, and he proved it for minimal
positive 2-spanning sets in Rr with r ≤ 4. For r ≥ 5, he gave a quadratic
bound on the size of minimal positive 2-spanning sets.
Theorem 10.2.3 (Marcus [71]). Let V ⊂ Rr be a minimal positive 2spanning set and n := |V |. Then
½
4r,
for r ≤ 4
n≤
r(r + 1)/2 + 5,
for r ≥ 5.
Marcus’ conjecture translates to the following conjecture on k-unneighborly
polytopes; see also Marcus [71].
Conjecture 10.2.4 (Marcus [71] [73]; disproved (Corollary 10.3.3)).
Let P be a k-unneighborly d-polytope, then
f0 (P ) ≥
If f0 =
2k
(d
2k−1
2k
(d + 1).
2k − 1
+ 1), then P is of type
(∆k−1 ⊕ ∆k−1 ) ∗ · · · ∗ (∆k−1 ⊕ ∆k−1 ).
173
For unneighborly polytopes, this conjecture asserts that the number of
vertices satisfies
f0 ≥ ⌈4(d + 1)/3⌉,
and the following gives a construction for polytopes that attain equality for
every d ≥ 2.
Indeed, what we describe here are whole classes of polytopes, as not
even the combinatorial type is determined by the description in one case.
However, all polytopes constructed are unneighborly, and the number of
vertices is the same for all polytopes in the same class.
(1) For d = 3ℓ − 1, we let Md be the set of polytopes obtained by taking
a join of ℓ-quadrilaterals.
Because the graph of the join of two polytopes is the join of the graphs of
the polytopes, every polytope in Md , for d = 3ℓ − 1, is unneighborly.
(2) For dimensions d = 3ℓ, we take a polytope P in Md−1 , that is, a join
of ℓ-quadrilaterals, and realize this polytope in the hyperplane xd = 0
in Rd . We then “attach” a quadrilateral along a common edge. That
is, we take two vertices (v0 , 0) ∈ Rd−1 × R and (v1 , 0) ∈ Rd−1 × R of P
that are connected by an edge, and take the convex hull of P and the
two points (v0 , 1) and (v1 , 1). Denote the set of all polytopes obtained
in this way by Md .
As the points {(v0 , 0), (v1 , 0), (v0 , 1), (v0 , 1)} form a quadrilateral, the resulting polytope is unneighborly. The combinatorial type of these polytopes
depends on the realization of the join of quadrilaterals. For example, for
d = 3 one may obtain a prism over a triangle, but also the polytope in
which one square of this prism is “broken” into two triangles.
(3) For dimensions d = 3ℓ+1, we again start with a join of ℓ-quadrilaterals,
that is, with a polytope P in Md−2 , and define Q as the vertex sum of
this polytope with a quadrilateral, that is, Q is of type (P, v0 ) ⊕ (¤, v1 ),
where v0 is a vertex of P and v1 is a vertex of ¤. Denote by Md the
set of all such polytopes.
Clearly, also every polytope in Md for d = 3ℓ + 1 is unneighborly.
A similar construction for polytopes that attain equality was given by
Marcus [71]. Indeed, one may construct polytopes in all dimensions such
2k
(d + 1)⌉ is tight for all d by modthat the integral bound of f0 ≥ ⌈ 2k−1
ifications, similar to those above, on joins of sums of simplices of higher
dimensions.
Comparing the functions s(d) and ⌈4(d + 1)/3⌉, we find that Marcus’
conjecture on 2-unneighborly polytopes fails for d = 36 (as claimed by
174
10.2. The Sizes of Minimal Positive Spanning Configurations
Marcus [73]) and for all d ≥ 39. For 4 ≤ d ≤ 29 and d = 31, 32 the
polytopes Md provide unneighborly polytopes on fewer vertices than Mani’s
examples (the polytopes in M2 and M3 have the same number of vertices
as the crosspolytopes). In all other dimensions (d = 30, 33, 34, 35, 37, 38)
both constructions give the same number of vertices.
It is not known what the extremal function for the number of vertices
of unneighborly polytopes is. By Theorem 10.2.3, a minimal positive 2spanning configuration in Rr on n vectors satisfies
n ≤ r(r + 1)/2 + 5.
According to this result, an unneighborly polytope has at least
p
f0 ≥ d + 2(d − 4)
√
vertices. Mani’s unneighborly polytopes have roughly d + 2 d vertices
(which is best possible for simplicial polytopes). So, for general
polytopes
√
√
the answer lies somewhere between the rough estimates d+ 2d and d+2 d.
It is not even known whether unneighborly polytopes on less than s(d)
vertices exist for “large” d. McMullen observed [79] that unneighborly polytopes on s(d) vertices that are not illuminated exist at least in dimensions
d ∈ {p2 + 1, p2 + 2, p2 + 3, p(p + 1) + 1, p(p + 1) + 2, p(p + 1) + 3}.
These can be obtained by “local” modifications of Mani’s illuminated polytopes, for example, by taking a join with a quadrilateral, a subdirect sum
with an edge, or a subdirect sum with a quadrilateral, similar to the modifications described above.
Before we construct counterexamples for all k ≥ 2, we look at classes of
polytopes for which the polytopes Md are extremal examples.
10.2.3
Special Classes that Satisfy Marcus’ Bound
We consider unneighborly polytopes that are unneighborly because of lowdimensional diagonals. That is, we fix a constant m and look at those
polytopes where every vertex lies on a diagonal through the relative interior
of a q-dimensional face with 2 ≤ q ≤ m.
The polytopes Md that we constructed in the last section are of this
type with m = 2 and much of the (false) intuition why one would believe
Conjecture 10.2.4 to be true is probably derived from these examples.
Definition 10.2.5 (Dimension of diagonals [31]). Let P be a d-polytope.
For 1 ≤ m ≤ d, an m-diagonal or a diagonal of dimension m of P is a segment [v, w], where v, w are vertices of P , such that the smallest face that
contains [v, w] is of dimension m.
175
The 1-diagonals of a polytope are the edges of the polytope and the
d-diagonals are the inner diagonals.
Similar to Marcus’ conjecture, one may ask for the minimum number of
vertices of a polytope such that every vertex lies on a diagonal of dimension
at most, or exactly, some fixed constant. For inner diagonals this number
is given by Mani’s result on illuminated polytopes [70]; see Theorem 9.1.4.
Theorem 10.2.6. Let P be d-polytope and fix 2 ≤ m ≤ d − 2. Suppose that
every vertex lies on a diagonal of dimension q with 2 ≤ q ≤ m. Then
¼
»
(m + 2)(d + 1)
.
f0 (P ) ≥
m+1
In particular, if m = 2, then f0 (P ) ≥ ⌈4(d + 1)/3⌉.
Proof. Because every vertex of P lies on a diagonal of dimension at most
m, there are faces F1 , . . . , Fℓ of P with the following properties:
(i) We have mi := dim Fi ≤ m.
(ii) The face Fi carries an mi -diagonal of P , that is, the polytope Fi has
an inner diagonal. This implies f0 (Fi ) ≥ mi + 2.
(iii) The polytope P is the convex hull of the vertices of the faces F1 , . . . Fℓ ,
that is, P = conv{F1 , . . . , Fℓ }.
But then
d≤ℓ−1+
ℓ
X
j=1
mi ≤ ℓ − 1 +
ℓ
X
j=1
m = ℓ(m + 1) − 1,
and we get that
¼
(m + 2)(d + 1)
.
f0 (P ) ≥
mi + 2ℓ = d + ℓ + 1 ≥
m
+
1
j=1
»
ℓ
X
For m = 2, this reads as f0 (P ) ≥ ⌈4(d + 1)/3⌉.
For m = d, this yields the trivial lower bound of f0 ≥ d + 2.
Another peculiar property of a join of quadrilaterals is that it can be
realized as a Lawrence polytope, that is, a polytope obtained from a series of Lawrence extensions; see Ziegler [118, Theorem and Definition 6.26].
These polytopes have the property that they have a centrally symmetric
Gale diagram. It follows from Marcus’ result on linearly k-spanning configurations [71] that every Lawrence polytope also satisfies the conjectured
bound of f0 ≥ ⌈4(d + 1)/3⌉.
176
10.3. Counterexamples to Marcus’ Conjecture
The joins of quadrilaterals are also the only polytopes with the maximum possible number of disjoint missing edges, by Theorem 5.4.14 and
Lemma 7.3.1.
Thus, the polytopes Md are indeed extremal examples for a number
of classes of polytopes, but as we have already seen not for the class of
unneighborly polytopes.
Theorem 10.2.6 implies that unneighborly polytopes that have less vertices than Mani’s illuminated ones must have high-dimensional diagonals.
10.3
Counterexamples to Marcus’ Conjecture
We now construct counterexamples to Conjectures 10.2.2 and 10.2.4 for all
k ≥ 2.
Theorem 10.3.1. Let k ≥ 2 and r ≥ 1 and write
¹ º µ¹ º ¹
º
¶
k
r
r+1
f (k, r) :=
+r+1 .
2
2
2
Then there is a minimal positive k-spanning configuration in Rr of size n,
where
(
f (k, r),
if k is even,
n=
¥ r+1 ¦
if k is odd.
f (k, r) + 2 + 1,
Proof. The construction depends on the parity of k, so we distinguish two
cases.
Case k = 2ℓ. We choose d depending on whether r is even or odd. If
r = 2p, we set d := p2 . Then
' &p '
&p
p
4p2 + 1 − 1
4p2 + 1 − 1
4p2
p−1<
≤
= p,
≤
2
2
2
that is, p = p(d).
If r = 2p + 1, we set d := p(p + 1). Then
&p
' &p
'
4p(p + 1) + 1 − 1
(2p + 1)2 − 1
p(d) =
=
= p.
2
2
Written uniformly, we have d = ⌊r/2⌋⌊(r + 1)/2⌋ and p(d) = ⌊r/2⌋.
Let A be a Gale diagram of I d . Then the size of A is
»
¼
¹ º¹
º
d
r
r+1
|A| = d + p(d) +
+1=
+ r + 1.
p(d)
2
2
177
Take every vector of this configuration ℓ times and denote the resulting
vector configuration by Aℓ . (On the polytope side, we have just taken the
wreath product of I d with an (ℓ − 1)-simplex [57].) The size n of Aℓ is
k
ℓ|A| =
2
µ¹ º ¹
º
¶
r
r+1
+ r + 1 = f (k, r).
2
2
It remains to show that Aℓ is a minimal positive k-spanning configuration.
Let H be an open halfspace of Rr . Since H contains at least 2 vectors
of A, it contains at least 2ℓ = k vectors of Aℓ . Thus Aℓ is positively kspanning.
Furthermore, if u ∈ Aℓ , then u is also contained in A. Now there is
an open halfspace H(u), such that H(u) contains exactly 2 vectors, one of
them being u. But then H(u) contains exactly 2ℓ vectors of Aℓ , one of them
being u. Thus Aℓ is a minimal positive k-spanning configuration.
Case k = 2ℓ + 1. We now construct configurations for k = 2ℓ + 1. Let
A and Aℓ be constructed as in the first case, and let {u1 , . . . , uq+1 }, where
q = ⌈d/p⌉, be those vectors of A that correspond to the stacked vertices of
I d.
For every i = 1, . . . , q + 1 add one copy of ui to Aℓ , and denote the
resulting vector configuration by Ãℓ . Then the size n of Ãℓ is
¹
º
¹
º
r+1
r+1
n = |Aℓ | + q + 1 = |Aℓ | +
+ 1 = f (k, r) +
+ 1.
2
2
It remains to show that Ãℓ is a minimal positive k-spanning configuration for Rr .
Every open halfspace contains at least 2 vectors of A, and at least one of
them is an element of {u1 , . . . , uq+1 }. Thus every open halfspace contains
at least 2ℓ + 1 vectors of Ãℓ . Consequently, Ãℓ is positively k-spanning.
If u is a vector of A\{u1 , . . . uq+1 }, then there is a vector ui ∈ {u1 , . . . , uq+1 }
and an open halfspace H(u, ui ) such that H(u, ui ) ∩ A = {u, ui }. Then
H(u, ui ) contains exactly 2ℓ+1 vectors of Ãℓ . If u is a vector in {u1 , . . . , uq+1 },
then there is a vector u ∈ A \ {u1 , . . . uq+1 } and an open halfspace H(u, ui )
such that H(u, ui ) ∩ A = {u, ui }. Again, H(u, ui ) contains exactly 2ℓ + 1
vectors of Ãℓ .
Thus, Ãℓ is a minimal positive k-spanning configuration.
Corollary 10.3.2. Conjecture 10.2.2 fails for all r ≥ 12 if k is even, and
for all r ≥ 18 if k is odd.
178
10.4. Upper Bound on the Size
Proof. Simple calculations show that for r ≥ 12 and k ≥ 2, the strict
inequality
f (k, r) > 2kr,
is satisfied, and for r ≥ 18 and k ≥ 3, the strict inequality
º
¹
r+1
+ 1 > 2kr
f (k, r) +
2
is satisfied. The statement follows.
Corollary 10.3.3. Conjecture 10.2.4 fails for all d with
(
f (k, r) − r − 1,
k even, r ≥ 12,
d=
f (k, r) − ⌊(1 − r)/2⌋ ,
k odd, r ≥ 18.
Based on the counterexamples one can also prove that Conjecture 10.2.4
indeed fails for all dimensions d ≥ d(r, k), for some function d(k, r).
Let P be a counterexample, that is, a polytope with a Gale diagram
as constructed in the proof of Theorem 10.3.1 for large r. If we start the
construction with I d , then P is a simplicial polytope.
Take the join of P with several sums of (k − 1)-simplices ∆k−1 ⊕ ∆k−1
and denote the resulting polytope by Q. To get the intermediate dimensions
we can “attach” a sum of two (k − 1)-simplices along an ℓ-simplex of P ,
which is a face of Q, with −1 ≤ ℓ ≤ 2k − 1, similar to the construction
of the polytopes Md . This is possible since P and ∆k−1 ⊕ ∆k−1 are both
simplicial polytopes, so both really have an ℓ-simplex face. The resulting
polytope may fail to be a counterexample only for “small” dimensions.
10.4
Upper Bound on the Size
As Marcus’ conjecture fails for all k ≥ 2, and there is no obvious substitute
for it, we formulate a more modest goal as the following problem; compare
with Problem 11.0.1 in the next chapter.
Problem 10.4.1. Prove an upper bound on the size of minimal positive
k-spanning configurations in Rr .
This was done by Marcus for the case k = 2 [71]; see Theorem 10.2.3
above. We add a bound on the size of minimal positive k-spanning configurations by proving an upper bound for all k ≥ 2.
179
Theorem 10.4.2. Let V ⊂ Rr be a minimal positive k-spanning configuration for k ≥ 2, and let n := |V |. Then
n ≤ kr(r + 1)k−1 .
Proof. Let P be a d-polytope that has V as a Gale diagram, where d =
n − r − 1. Then P is k-unneighborly, that is, every vertex is contained in
an empty (k − 1)-simplex.
By Theorem 7.2.4, the number of empty (k − 1)-simplices is bounded
by r(r + 1)k−1 . Thus the number n of vertices of P is bounded by
n ≤ f0 (∆k−1 )r(r + 1)k−1 = kr(r + 1)k−1 ,
as claimed.
As we have seen in Theorem 10.3.1, there exist, for fixed k, minimal
positive k-spanning configurations of size Θ(r2 ). Thus there is still a large
gap to the upper bound of O(rk ).
We formulate and prove the corresponding result for k-unneighborly
polytopes.
Corollary 10.4.3. Let P be a k-unneighborly d-polytope. Then P has at
least
p
f0 ≥ d + k (d + 1)/k
vertices.
Proof. Let P be k-unneighborly and write f0 = d + r + 1. Then
d + r + 1 ≤ kr(r + 1)k−1 ,
by applying Theorem 10.4.2 to a Gale diagram of P .
Thus we have
(d + 1)/k ≤ (r + 1)k .
p
This implies that r + 1 ≥ k (d + 1)/k and we get
p
f0 = d + r + 1 ≥ d + k (d + 1)/k,
as claimed.
180
Chapter 11
Positive Spanning Sets in
Oriented Matroids
In this chapter we generalize the result of Chapter 10 on the size of minimal
positive k-spanning configurations to oriented matroids.
Gale duality is at the core of the interrelation between minimal positive
k-spanning configurations and k-unneighborly polytopes, as discussed in
Chapter 10. The setup and questions generalize to oriented matroids: We
speak of totally cyclic oriented matroids instead of positive spanning sets,
of acyclic oriented matroids instead of affine point sets, and of oriented
matroid duality instead of Gale duality.
In this setting the following problem had been posed by Bienia & Las
Vergnas [23, Exercise 9.35(iv)*].
Problem 11.0.1 (Bienia & Las Vergnas [23, Exercise 9.35(iv)*]).
Let M be a totally cyclic oriented matroid on a set E.
A subset A ⊆ E such that convM (A) = E is called a positive spanning
set. Given an integer k ≥ 1, a positive k-spanning set is a subset A ⊆ E
such that A \ S is a positive spanning set for all S ⊆ A with |S| < k.
Find a function f (k, r(M)) such that for every positive k-spanning set
A there is a positive k-spanning set B ⊆ A with |B| ≤ f (k, r(M)).
In this chapter we derive such a function from Perles’ Skeleton Theorem.
Bienia and Las Vergnas [15] generalized the Blumenthal–Robinson Theorem (Theorem 10.2.1) to oriented matroids. Consequently, for k = 1 the
function f (k, r(M)) is given by 2r(M) and this bound cannot be improved.
181
11. Positive Spanning Sets in Oriented Matroids
We have at least two ways to obtain a function f (k, r(M)) for k ≥ 2 by
applying Perles’ Skeleton Theorem.
We may use the bound we proved in Theorem 8.11.1 for graded relatively
complemented lattices. The face lattice (covector lattice) of an oriented
matroid is a graded relatively complemented lattice [23, Theorem 4.1.14,
Corollary 4.1.16].
To obtain the better bound proved for geometrically realizable totally
cyclic oriented matroids in Theorem 10.4.2 we invoke the topological representation theorem for oriented matroids by Folkman & Lawrence [42],
Edmonds & Mandel [39], and Lawrence [68]. This is a deep theorem in the
theory of oriented matroids, which establishes the hard part of the following
chain of inclusions,
{polytopes} ⊆ {matroid polytopes} ⊆ {strong PL spheres},
namely, that matroid polytopes can be represented by strong PL spheres.
We then apply Theorem 8.4.4.
11.1
Oriented Matroids
We state in this section the basic definitions for oriented matroids. The
theory of oriented matroids is a broad field, and many interesting things
can be said about them. We keep this part brief and introduce just the
right amount of notions to be able to state Bienia & Las Vergnas’ problem
(Problem 11.0.1). Notation and definitions are taken from [23, Chapter 3],
and we refer to [23] for more (indeed, much more) on oriented matroids.
A signed set X is a set X together with a partition (X + , X − ) into the
positive elements X + and the negative elements X − . The set X is the
support of X. A signed set is positive if X − = ∅, and it is negative if
X + = ∅. We write ∅ for the signed set (∅, ∅).
Let E be a finite set. A signed subset of E is a signed set with support in
E. As usual, we identify a signed subset of E with an element in {+, 0, −}E .
In particular, if E = [n] we write a signed subset as a sign vector of length n.
For an element e of a signed subset X ⊆ E we define X(e) := 1, if
e ∈ X + , X(e) := −1, if e ∈ X − , and X(e) := 0, if e ∈ E \ X.
The opposite of a signed set X is the signed set −X with (−X)+ = X −
and (−X)− = X + . If C is a collection of signed sets we write − C for the
collection {−X : X ∈ C}.
We define oriented matroids in terms of circuit systems. As for matroids
there are many axiom systems for oriented matroids, all meticulously detailed in [23, Chapter 3].
182
11.2. Oriented Matroids, Polytopes, and Duality
Definition 11.1.1 (Oriented matroid). An oriented matroid M is a pair
(E, C) of a finite set E and a collection C of signed subsets of E, called the
circuits of M, that satisfies the following axioms:
(i) ∅ ∈
/ C,
(ii) C = − C,
(iii) for all X, Y ∈ C, if X ⊆ Y , then X = Y or X = −Y ,
(iv) for all X, Y ∈ C, X 6= −Y and e ∈ X + ∩ Y − there is a Z ∈ C such
that
Z + ⊆ (X + ∪ Y + ) \ {e} and Z − ⊆ (X − ∪ Y − ) \ {e}.
We define the composition X ◦ Y of signed sets X, Y by
(X ◦ Y )+ = X + ∪ (Y + \ X − ) and (X ◦ Y )− = X − ∪ (Y − \ X + ).
A vector of an oriented matroid is any composition of circuits.
The signed sets X, Y are said to be orthogonal if either X ∩ Y = ∅ or if
there are e, f ∈ X ∩ Y such that X(e)Y (e) = −X(f )Y (f ).
The following proposition establishes the existence of a dual oriented
matroid.
Proposition 11.1.2 (Bland & Las Vergnas [24]; see [23, Proposition
3.4.1]). Let M be an oriented matroid E with circuits C.
(i) There is a unique signature C ∗ of the cocircuits of the underlying matroid of M such that for all X ∈ C and Y ∈ C ∗ we have that X and
Y are orthogonal.
(ii) The collection C ∗ is the set of circuits of an oriented matroid on E,
called the dual of M and denoted by M∗ .
(iii) We have (M∗ )∗ = M, that is, the dual of the dual is the oriented
matroid we started with.
The circuits and vectors of M∗ are called the cocircuits and covectors
of M, respectively.
11.2
Oriented Matroids, Polytopes, and Duality
We relate polytopes to oriented matroids and mention the relationship between Gale duality and oriented matroid duality. This establishes a connection to the material of the previous chapter.
183
11.2.1
From Polytopes to Oriented Matroids
Given a d-polytope P , we define an oriented matroid M := M(P ) of rank
r = d + 1 that encodes the combinatorial structure of P in the following
way; see [23, p. 164] and also [118, Chapter 6], where oriented matroids are
discussed with the primary intention of applying them to polytope theory.
Let P ⊂ Rd have vertices V(P ) = {v1 , . . . , vn }, and let f (x) = cT x + d
be an affine function on Rd .
We record the signs of the values of f on the vertices of P as the sign
vector sgn(f (v1 ), . . . , f (vn )). These sign vectors constitute the collection of
covectors of an oriented matroid M(P ).
We get the covector lattice of M(P ) if we order the sign vectors according to the following partial order: Take the natural partial order on
{+, −, 0} given by the two relations 0 < + and 0 < −. The partial order
on the sign vectors then is given by componentwise extension of this order.
For every proper nonempty face F of P we find a supporting hyperplane given by an affine function f (x) = cT x + d with f (v) = 0 for vertices
v on F and f (v) > 0 for all vertices v of P not on F . The sign vector sgn(f (v1 ), f (v2 ), . . . , f (vn )) does not depend on the exact choice of f
but only on the combinatorial structure of P . Obviously, it contains only
positive or zero entries. That is, the covectors corresponding to faces of
P are all contained in the interval F = [0, +], where 0 = (0, . . . , 0) and
+ = (+, . . . , +). The top element in this interval F corresponds to the
empty set, while the bottom element corresponds to P itself.
Thus, the interval F is the face lattice of the polar of P , or in other
words F op = L(P ). The lattice F op is called the Las Vergnas face lattice
of the oriented matroid M.
11.2.2
From Gale Diagrams to Oriented Matroids
Let G be a Gale diagram of a d-polytope P . Then G lies in Rm , where
m = f0 (P ) − d − 1. The vector configuration G gives rise to an oriented
matroid M* = M* (P ) of rank r∗ = m in the following way: The sign
vectors of linear dependences among g1 , . . . , gn form the circuit system of
an oriented matroid.
In particular, positive dependences correspond to positive covectors in
M, that is, to faces of the polytope P . Most importantly, M* is the oriented
matroid dual to M.
According to Ziegler [118, p. 183], the connection between oriented matroid duality and Gale duality was first observed by McMullen [78]. It was
worked out by Sturmfels [110].
184
11.3. Positive Spanning Sets in Oriented Matroids
11.3
Positive Spanning Sets in Oriented Matroids
When talking about polytopes and Gale diagrams of polytopes, we are talking about objects with the following properties: The vertices of a polytope
form an affine point set, and the vectors of a Gale diagram form a positive
spanning vector configuration. These two properties are translated into the
world of oriented matroids by the notions of “acyclic” and “totally cyclic.”
Definition 11.3.1 (Acyclic, totally cyclic, positive spanning set).
Let M be an oriented matroid on a set E and denote by C(M) the circuits
of M.
The oriented matroid M is acyclic if it does not contain a positive
circuit. It is totally cyclic if every element is contained in a positive circuit.
For A ⊆ E, we define the convex hull of A in M by
convM (A) := A ∪ {e ∈ E \ A : there is X ∈ C(M )
such that e ∈ X − and X + ⊆ A}.
If M is totally cyclic, we call a subset A ⊆ E with convM (A) = E a positive
spanning set.
Bienia & Las Vergnas [15] have shown the generalization of the theorem
by Blumenthal & Robinson, Theorem 10.2.1, in this setting.
Theorem 11.3.2 (Bienia & Las Vergnas [15]). Let M be a totally cyclic
oriented matroid.
(i) A set A ⊆ E is a positive spanning set if and only if rM (A) = r(M)
and A is a union of positive circuits.
(ii) For every positive spanning set A there is a positive spanning set B ⊆
A such
≤ 2r(M). Furthermore, |B| = 2r if and only if
Sr that |B|
′
B = i=1 {ei , ei }, where {e1 , . . . , er } is a basis of M and {ei , e′i } is a
positive circuit for i = 1, . . . , r.
Definition 11.3.3 (Positive k-spanning set). Given an integer k ≥ 1,
a positive k-spanning set is a subset A ⊆ E such that A \ S is a positive
spanning set for all S ⊆ A with |S| < k.
Let M be a totally cyclic oriented matroid of rank r. Problem 11.0.1
by Bienia & Las Vergnas is to find a function f (k, r) such that for every
positive k-spanning set A ⊆ E there is a positive k-spanning set B ⊆ A
with |B| ≤ f (k, r).
185
Given a positive k-spanning set A there is a minimal, that is, inclusionminimal, positive k-spanning set B ⊆ A. So, the above question can be
answered by finding a bound on the size of minimal positive k-spanning
sets.
Clearly, a positive k-spanning set is positively spanning, so B is a union
of positive circuits, by Theorem 11.3.2 (i). By definition, M \(E \B) then is
totally cyclic. Thus we can assume that E is a minimal positive k-spanning
set for M. For k ≥ 2 this implies that M* is a matroid polytope, that is,
all one-element subsets of E are vertices [23, Chapter 9].
11.4
The Topological Representation Theorem
The topological representation theorem for oriented matroids due to Folkman & Lawrence [42], Edmonds & Mandel [39], and Lawrence [68] asserts
that every oriented matroid arises as the oriented matroid of an arrangement of pseudospheres; see for example the exposition by Björner [21].
For us, the following is the relevant part of the topological representation
theorem for oriented matroids:
Theorem 11.4.1 (see [23, Theorem 4.3.5]). Let M be an acyclic oriented matroid of rank r. The Las Vergnas lattice of M is isomorphic to the
face lattice of a strong PL (r − 2)-sphere.
11.5
Upper Bound on Minimal Positive k-Spanning
Sets
We now apply Perles’ Skeleton Theorem to Problem 11.0.1 by Bienia & Las
Vergnas on positive k-spanning sets in oriented matroids.
We define k-unneighborly for strong PL spheres in the same manner as
for polytopes.
Definition 11.5.1 (k-unneighborly). Let P be a strong PL sphere.
Then P is called k-unneighborly , if (a) it is (k − 1)-neighborly, that is,
every set of k vertices is the set of vertices of a (k − 2)-cell, and (b) every
vertex of P lies in an empty (k − 1)-simplex.
Lemma 11.5.2. Let M be a totally cyclic oriented matroid on set E, and
suppose that E is a minimal positive k-spanning set for k ≥ 2. Denote by
M* the corresponding matroid polytope.
If P is a strong PL sphere isomorphic to the Las Vergnas face lattice of
*
M , then P is k-unneighborly.
186
11.5. Upper Bound on Minimal Positive k-Spanning Sets
Proof. Let F ⊆ E be a set of size at most k − 1. Then E \ F is positively
spanning in M, that is, it is the support of a positive vector. Then F is
the zero-set of a positive covector of M* , that is, of a face of M* .
For every v ∈ E, there is a set S ⊂ E with |S| = k and v ∈ S, such that
E \ S is not positive spanning in M. Then S is not a face of M* .
We can now state and prove the upper bound on minimal positive kspanning sets. This solves Problem 11.0.1. As in the realizable case there
is no indication why the bound proved should by asymptotically optimal.
Theorem 11.5.3. Let M be a totally cyclic oriented matroid of rank r and
suppose that E is a minimal positive k-spanning set of M for k ≥ 2. Then
|E| ≤ kr(r + 1)k−1 .
Proof. The matroid polytope M∗ is isomorphic to a strong PL (d − 1)sphere P for d := |E| − r − 1 on d + r + 1 vertices. The sphere P is
k-unneighborly by Lemma 11.5.2.
Apply Theorem 8.4.4 to P . Then the number of vertices of P , which
equals |E| as k ≥ 2, is bounded by kr(r + 1)k−1 .
187
Appendix
Appendix A
Poset of Structures
Figure A.1 summarizes the inclusion relations among structures that appear
in this thesis; see also Figures 8.1 and 4.1. Structures that are central for
this thesis are indicated by a bold font.
Some of the inclusions are trivial by definition, as strong PL spheres
and strongly regular cell complexes are of course regular cell complexes,
and the face poset of any structure is a poset.
The association of an oriented matroid to a polytope is discussed in
Section 11.2. That matroid polytopes are isomorphic to strong PL spheres
is a consequence of the Topological Representation Theorem for oriented
matroids; see Section 11.4.
Strongly regular cell decompositions of manifolds are graph manifolds,
that is, the collection of the graphs of the cells of such a cell complex form
a graph manifold; see Chapter 4, in particular also the more detailed figure
in Section 4.3.
The face poset of a weak graph manifold is a graded relatively complemented lattice. Indeed, such a face poset satisfies the diamond property.
The inclusion then follows from Björner’s criterion for relatively complementedness; see Theorem 1.2.2.
Finally, graded relatively complemented lattices are pyramidally perfect.
This was shown in Section 8.8.
191
A. Poset of Structures
posets
pyramidally perfect
lattices
(Chapter 8)
graded relatively
complemented lattices
(Chapter 8)
weak graph manifolds
graph manifolds
(Chapter 4)
regular cell complexes
strongly regular cell decompositions of manifolds
cellular PL spheres
strong PL spheres
(Chapters 8, 11)
matroid polytopes
(Chapter 11)
polytopes
(Chapters 2, 3, 5, 6, 7, 9,
and 10)
Figure A.1: Inclusion relations among the structures that appear in this thesis.
192
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List of Symbols
(P1 , F ) ⊕ (P2 , G)
the subdirect sum of P1 and P2 with respect to the
faces F and G
[n]
the set of all natural numbers from 1 to n
[x, y]
the interval between elements x, y of a poset
A(L)
the atoms of the lattice L
A(x)
C
the atoms in a lattice below the element x
a regular cell complex
C[U ]
subset of the cells of the regular cell complex C
induced by the set U
astC (F )
the antistar of the regular cell complex C at the cell
F
starC (F )
the star of the regular cell complex C at the cell F
∆d
the d-dimensional standard simplex
∅
the empty set
L(P )
F(C)
F(C)
b
F(C)
γ(P )
the face lattice of the polytope P
face poset of the regular cell complex C
the face poset of the polytopal complex C
the augmented face poset of a regular cell complex
the quantity f0 − d − 1 of the strong PL (d − 1)sphere P
203
List of Symbols
G (n, j1 , k1 , . . . , jm , km ) the polytope ∆n−1 ∗(∆j1 ⊕ ∆k1 )∗· · ·∗(∆jm ⊕ ∆km )
for parameters n, m ≥ 0 and j1 , k1 , . . . , jm , km ≥ 1
G (n, m)
the polytope ∆n−1 ∗ |¤ ∗ ·{z
· · ∗ ¤} for n, m ≥ 0
m times
int P
the set of interior points of the polytope P
∨
the join of two elements in a join-semilattice
κk,ℓ (d)
the minimum connectivity of (k, ℓ)-incidence graphs
in dimension d
linkC (F )
the link of the regular cell complex C at the cell F
K
a geometric simplicial complex
M /F
face figure of the graph manifold M at F
∧
the meet of two elements in a meet-semilattice
1̂
the maximal element of a poset that has a 1̂
¤
a 2-dimensional cube
◦
σ
the interior of a subset σ of a topological space
V(C)
the vertices of the polytopal complex C
M
a face structure
V(M)
the vertices of the face structure M
V(P )
the set of vertices of the regular cell complex C
0
the column vector in which every entry is zero
V(P )
0̂
Ck
the set of vertices of the polytope P
the minimal element of a poset that has a 0̂
the simple cycle on k edges
k
the graph on 2k vertices with exactly k disjoint
edges
f0 (C)
the number of vertices of the regular cell complex
C
E
G(C)
G1 ∪· G2
G 1 ∗ G2
Gc (P )
the graph of the regular cell complex C
the disjoint union of the graphs G1 and G2 (under
the assumption that V (G1 ) ∩ V (G2 ) = ∅)
the join of the graphs G1 and G2
the graph of P directed according to the linear
function c
204
List of Symbols
Gk (C)
Gk (M)
Gk (P )
the (k, k + 1)-incidence graph of the regular cell
complex C
the (k, k+1)-incidence graph of the graph manifold
M
the (k, k + 1)-incidence graph of the polytope P
Gcofacet (P )
the hypergraph of facet complements of the polytope P
Gk,ℓ
the (k, ℓ)-incidence graph of the polytope P
Gk,ℓ (M)
the (k, ℓ)-incidence graph of the graph manifold M
k(d)
the minimal linkedness of d-polytopes
k(d, γ)
the minimal linkedness of polytope in P γd
K(d, γ)
the maximal linkedness of a d-polytope on d+γ +1
vertices
k(G)
the largest integer k such that the graph G is klinked
k(P )
the minimal linkedness of the graph of the polytope
P
kS (d)
the minimal linkedness of simlicial d-polytopes
kS (d, γ)
the minimal linkedness of polytopes in S γd
M/F
the quotient of the empty face M at the cell F
P/F
the face figure of the strong PL sphere P at the
cell F
Pk
the path on k edges
P1 ∗ P2
the join of the polytopes P1 and P2
P1 ⊕ P2
the direct sum of the polytopes P1 and P2
P1 × P2
the cartesian product of the polytopes P1 and P2
T (P )
the (deep) truncation of the truncatable polytope
P
aff(V )
the affine hull of the set V ⊆ Rd
astC (F )
B(A)
B(a, . . . , an )
the closed antistar of the face F in the polytopal
complex C
the boolean lattice on the set of atoms A
the boolean lattice on atoms {a1 , . . . , an }
205
List of Symbols
bipyr P
the bipyramid over P
B(P )
the boundary complex of the polytope P
C
a polytopal complex
C3∆
the octahedron
Cd
the d-dimensional standard cube
Cd∆
the d-dimensional standard crosspolytope
cone(V )
the conical hull of the set V ⊆ Rd
conv(V )
the convex hull of the set V ⊆ Rd
dim(P )
the dimension of the polytope P
E
the edges of a graph
E(G)
the edges of the graph G
F3
the face of the polar polytope that corresponds to
the face F
f (P )
the f -vector of the polytope P
G
the complement of the graph G
G
a finite graph
γ(P )
the quantity f0 − d − 1
h
the h-vector of a simplicial polytope
I
the unit interval
κ(G)
the connectivity of the graph G
Km,n
the complete bipartite graph on m + n vertices
¡ ¢
) on n vertices
the complete graph G = ([n], [n]
2
Kn
linkC (F )
the link of the face F in the polytopal complex C
maxc (P )
the set of maxima of the polytope P with respect
to the linear function c
minc (P )
the set of minima of the polytope P with respect
to the linear function c
N
the natural numbers N = {0, 1, 2, . . .}
n
a natural number
N (v)
neighbors of the vertex v in a graph
P
a polytope
P γd
the set of d-polytopes on d + γ + 1 vertices
206
List of Symbols
P/F
the face figure of P at the face F
P∆
the polar of the polytope P
P/v
the vertex figure of P at the vertex v
pyrk (P )
the k-fold pyramid over P
pyra (P )
the pyramid over P with apex a
R
the real numbers
rk(L)
the rank of the graded poset L
rk(x)
the rank of the element x in a graded poset
S γd
the set of simplicial d-polytopes on d+γ +1 vertices
starC (F )
the closed star of the face F in the polytopal complex C
skelk (C)
the k-skeleton of the polytopal complex C
V
the vertices of a graph or polytope
V (G)
the vertices of the graph G
207
Index
acyclic . . . . . . . . . . . . . . . . . . . . . . . 185
affine Gale diagrams . . . . . . . . . 166
affine hull . . . . . . . . . . . . . . . . . . . . . 13
affine point set . . . . . . . . . . . . . . . 181
antistar
face structure . . . . . . . . . . . . . 60
antisymmetry . . . . . . . . . . . . . . . . . 10
apex . . . . . . . . . . . . . . . . . . . . . 19, 124
apex of a pyramid . . . . . . . . . . . . . 18
arrangement of
pseudospheres . . . . . . . . 186
Athanasiadis’ conjecture . . vii, 53,
54 – 55
atom-induced . . . . . . . . . . . . . . . . 139
augmented face poset . . . . . . . . 124
bipyramid . . . . . . . . . . . . . . . . . . . . . 19
Blumenthal–Robinson
Theorem . . . 169, 172, 173,
185
classical . . . . . . . . . . . . . . . . . 172
for oriented
matroids . . . . . . . . 181, 185
large simplex face . . . . . . . . . 85
Shephard’s proof . . . 172 – 173
boolean . . . . . . . . . . . . . . . . . . . . . . . 12
boolean intervals . . . . . . . . . . . . . 137
boolean lattice . . . . . . . . . . . . . . . . 12
bound is not tight . . . . . . . . . . . . . 81
boundary complex. . . . . . . . . . . . .19
Brückner sphere . . . . . . . . . . . . . . 121
bad edge . . . . . . . . . . . . . . . . . . . . . . 68
bad vertex . . . . . . . . . . . . . . . . . . . . 43
Balinski’s theorem . . . . . . . . . . . . . . 1
for graph manifolds. . . . . . . .63
for polytopes . . . . . . . . . . . . . . 38
generalizations of . . . . . . viii, 1
is best possible . . . . . . . . . . . . 41
barycentric coordinates. . . . . . .123
base . . . . . . . . . . . . . . . . . . . . . . 19, 124
base of a pyramid . . . . . . . . . . . . . 18
bipartite graph . . . . . . . . . . . . . . . . . 8
cellular PL ball . . . . . . . . . . . . . . 123
cellular PL sphere . . . . . . . . . . . 123
classification of polytopes on d + 2
vertices . . . . . . . . . 3, 90, 92
combinatorially isomorphic polytopal complexes . . . . . . . 19
combinatorially isomorphic polytopes . . . . . . . . . . . . . . . . . . 16
complement graph. . . . . . . . . . . . .10
complemented . . . . . . . . . . . . . . . . . 12
complete graph . . . . . . . . . . . . . . . . . 8
209
Index
concatenation
of paths . . . . . . . . . . . . . . . . . . . . 8
of strong chains . . . . . . . . . . . 61
of strong walks . . . . . . . . . . . . 62
of walks . . . . . . . . . . . . . . . . . . . 62
cone . . . . . . . . . . . . . see conical hull
conical hull. . . . . . . . . . . . . . . . . . . .13
conjecture
by Athanasiadis . . . . . . . . . . see
Athanasiadis’ conjecture
by Kalai,
Kleinschmidt & Lee . . 119
by Marcus . . . . . . see Marcus’
conjecture
connected graph . . . . . . . . . . . . . . . . 9
connectivity
and linkages . . . . . . . 75, 77, 81
of a graph . . . . . . . . . . . . . . . . . . 9
of graph manifold skeleta . . 64
of incidence graphs . . . . . . . . vii
convex hull . . . . . . . . . . . . . . . 13, 185
covector lattice . . . . . . . . . . 182, 184
cubical polytopes . . . . . . . . . . . . . . 89
cyclic polytopes . . . . . . . . . . . . . . 120
number of empty
simplices in . . . . . . . . . . 120
direct sum . . . . . . . . . . . . . . . . . . . . 19
disjoint empty faces . . . . . . . . . . 115
bound is tight . . . . . . . . . . . . 115
disjoint union of graphs . . . . . . 116
d-polytopal . . . . . . . . . . . . . . . . . . . . 48
d-simplex . . . . . . . . . . . . . . . . . . . . . 16
d-simplicial graph . . . . . . . . . . . . . 80
edges . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
elementary graphs. . . . . . . . . . . .116
empty face . . . . . . . . . . . . . . 109, 126
empty faces . . . . . . . . . . . . . . . . . . 111
flat empty faces . . . . . . . . . . 111
in regular cell complexes . 125
quotient of . . . . . . . . . . . . . . . 111
empty k-faces . . . . . . . . . . . . . . . . 111
flat empty k-faces . . . . . . . . 111
empty k-pyramids . . . . . . . . . . . . 111
empty k-simplices . . . . . . . . . . . . 111
empty pyramid . . . . . . . . . . . . . . .126
empty pyramids . . . . . . . . . . . . . . 111
in lattices . . . . . . . . . . . . . . . . 142
in quotient . . . . . . . . . 126 – 128
empty simplex . . . . . . . . . . . . . . . 126
empty simplices . . . . . . . . . 111, 186
bound on the number of . . 119
empty vertices . . . . . . . . . . . . . . . 173
endpoints . . . . . . . . . . . . . . . . . . . . . . 7
Erdős-Rado Sunflower
Lemma . . . . . . . . . . 115, 119
erroneous answer to Mani’s problem . . . . . . . . . . . . . . . . . . 163
d-graph complex . . . . . . . . . . . . . . 57
d-graph manifold . . . . . . . . . . . . . . 56
d-graph manifold with boundary
57
d-pseudo-graph manifold . . . . . . 57
d-crosspolytope . . . . . . . . . . . . . . . 16
d-cube . . . . . . . . . . . . . . . . . . . . . . . . 16
definitions of kernels
coincide . . . . . . . . . . . . . . 140
d-face structure . . . . . . . . . . . . . . . 56
diamond property. . . . .12, 91, 125
versus relatively
complemented . . . . . . . . . 12
dimension . . . . . . . . . . . . . . . . . . . . . 19
face structure
strongly connected . . . . . . . . 56
face figure . . . . . . . . . . . . . . . . . 15, 58
of a strong PL sphere . . . . 125
face lattice . . . . . . . . . . . . . . . . . . . . 15
face poset . . . . . . . . . . . . . . . . . . . . . 19
210
Index
face poset of a polytope . . . . . . . 14
face structure . . . . . . . . . . . . . . . . . 56
antistar . . . . . . . . . . . . . . . . . . . 60
closed under taking faces . . 56
edges . . . . . . . . . . . . . . . . . . . . . . 56
faces . . . . . . . . . . . . . . . . . . . . . . 56
facets . . . . . . . . . . . . . . . . . . . . . 56
intersection property . . . . . . 56
link . . . . . . . . . . . . . . . . . . . . . . . 60
pure . . . . . . . . . . . . . . . . . . . . . . .56
ridges . . . . . . . . . . . . . . . . . . . . . 56
star . . . . . . . . . . . . . . . . . . . . . . . 60
strong chain . . . . . . . . . . . . . . . 56
vertices. . . . . . . . . . . . . . . . . . . .56
flat . . . . . . . . . . . . . . . . . . . . . . . . . . 109
flat embedding . . . . . . . . . . . . . . . 123
flat empty faces . . . . . . . . . . . . . . 111
in quotient . . . . . . . . . . . . . . . 111
flat empty k-faces . . . . . . . . . . . . 111
free vertex . . . . . . . . . . . . . . . . . . . . 82
f -vector . . . . . . . . . . . . . . . . . . . . . . . 16
bipartite. . . . . . . . . . . . . . . . . . . .8
complement . . . . . . . . . . . . . . . 10
complete . . . . . . . . . . . . . . . . . . . 8
connected . . . . . . . . . . . . . . . . . . 9
connectivity . . . . . . . . . . . . . . . . 9
cycle . . . . . . . . . . . . . . . . . . . . . . . 9
edges . . . . . . . . . . . . . . . . . . . . . . . 7
independent paths . . . . . . . . . . 8
induced subgraph. . . . . . . . . . .8
inner vertices of a path . . . . . 8
isomorphic . . . . . . . . . . . . . . . . . 7
join . . . . . . . . . . . . . . . . . . . . . . . . 9
k-connected . . . . . . . . . . . . . . . . 9
length of a cycle . . . . . . . . . . . . 9
length of a walk . . . . . . . . . . . . 8
minor . . . . . . . . . . . . . . . . . . . . . 10
neighbors . . . . . . . . . . . . . . . . . . . 8
path . . . . . . . . . . . . . . . . . . . . . . . . 8
path concatenation . . . . . . . . . 8
path length . . . . . . . . . . . . . . . . . 8
polytopal . . . . . . . . . . . . . . . . . . 48
polytopality . . . . . . . . . . . . . . . 48
subgraph . . . . . . . . . . . . . . . . . . . 8
subpath . . . . . . . . . . . . . . . . . . . . 8
vertices . . . . . . . . . . . . . . . . . . . . . 7
walk . . . . . . . . . . . . . . . . . . . . . . . . 8
graph complex . . . . . . . . . . . . . . . . 57
graph manifold . . . . . . . . . . . . . . . . 56
(k, ℓ)-incidence graph . . . . . . 64
bad vertex . . . . . . . . . . . . . . . . 68
connecting walk . . . . . . . . . . . 66
dual graph . . . . . . . . . . . . . . . . 64
edge figure . . . . . . . . . . . . . . . . 58
face figure . . . . . . . . . . . . . . . . . 58
face numbers . . . . . . . . . . . . . . 60
good edge . . . . . . . . . . . . . . . . . 68
good vertex . . . . . . . . . . . . . . . 68
good walk . . . . . . . . . . . . . 66, 69
strong cycle . . . . . . . . . . . . . . . 65
strong walk. . . . . . . . . . . . . . . .61
vertex figure. . . . . . . . . . . . . . .58
Gale diagrams. . . . . . . . . . . . . . . . .17
example . . . . . . . . . . . . . . . . . . . 17
geometry of . . . . . . . . . . . . . . 116
main theorem of . . . . . . . . . . . 17
oriented matroid of . . . . . . . 184
Gale duality . . . . . . . . . . . . . . . . . 181
and oriented
matroid duality . . . . . . 183
Gallivan’s examples . . . . . . . . 76, 85
γ . . . . . . . . . . . . . . . . . . . . . . . . . 16, 138
generic . . . . . . . . . . . . . . . . . . . . . . . . 38
geometric lattices . . . . . . 121 – 122
geometric realization . . . . . . . . . 123
geometric simplicial complex . 122
geometries of rank 3
on 4 points . . . . . . . . . . . 142
good vertex . . . . . . . . . . . . . . . . . . . 43
graph . . . . . . . . . . . . . . . . . . . . . . . . . . 7
211
Index
graph manifold with boundary 57
g-vector . . . . . . . . . . . . . . . . . . . . . . . 16
of empty pyramids . . 127, 145
of geometric empty faces . 127
intersection property
face structure . . . . . . . . . . . . . 56
intersection property
polytopal complex . . . . . . . . . 19
regular cell complex . . . . . . . 54
isomorphic graphs . . . . . . . . . . . . . . 7
isomorphism of posets . . . . . . . . . 11
Hadwiger’s problem on illuminated
polytopes . . . . . . . . . . . . 155
Mani’s counterexamples . . 157
Handbook of Discrete and Computational Geometry . . 75
Hirsch conjecture . . . . . . . . . . . . . . . 1
h-vector . . . . . . . . . . . . . . . . . . . . . . . 16
join of two graphs . . . . . . . . . . . . . . 9
join of two polytopes . . . . . . . . . . 18
combinatorially . . . . . . . . . . . . 18
geometrically . . . . . . . . . . . . . . 18
join-semilattice . . . . . . . . . . . . . . . . 11
coatom . . . . . . . . . . . . . . . . . . . . 12
illuminate itself . . . . . . . . . . . . . . 164
illuminated. . . . . . . . . . . . . . . . . . .158
illuminated polytopes
Hadwiger’s problem . . . . . . 155
Mani polytope . . . . . . . . . . . 158
Mani’s counterexamples . 157,
160
Mani’s problem . . . . . . . . . . 157
Mani’s theorem . . . . . . . . . . 159
minimum number
of vertices . . . . . . . 158, 159
some examples . . . . . . . . . . . 162
incidence graph . . . . . . . . . . . . . . . 54
inclusion relations among face structures . . . . . . . . . . . . . . . . . . 57
inclusion relations among regular
cell complexes . . . . . . . . 124
independent paths . . . . . . . . . . . . . . 8
induced subcomplex . . . . . 103, 125
induced subgraph . . . . . . . . . . . . . . 8
inner diagonal . . . . . . . . . . . . . . . . 158
in simple polytopes . . . . . . .158
simpliciality . . . . . . . . . . . . . . 158
upper bound theorem by Bremner & Klee . . . . . . . . . . . 158
inner vertices of a path . . . . . . . . . 8
interdependence of chapters . . . . 2
interior points . . . . . . . . . . . . . . . . . 14
intersection
empty face with face . . . . . 126
Kalai’s proof of Perles’ Skeleton
Theorem . . . . . . . . . . . . . 115
k-connected . . . . . . . . . . . . . . . . . . . . 9
kernels of polytopes
on d + 2 vertices . . . . . . . . . 116
on d + 3 vertices . . . . . . . . . 116
k-flat. . . . . . . . . . . . . . . . . . . . . . . . . .13
skew . . . . . . . . . . . . . . . . . . . . . . 13
k-kernel . . . . . . . . . . . . . . . . . . . . . . 104
(k, ℓ)-incidence graph . . . . . . . . . .41
k-linked . . . . . . . . . . . . . . . . . . . 77, 77
k-skeleton . . . . . . . . . . . . . . . . 20, 139
k-unneighborly
PL sphere . . . . . . . . . . . . . . . 186
polytope . . . . . . . . . . . . . . . . . 171
large simplex face . . . . . . . . . . . . . 85
Las Vergnas face lattice . . . . . . 184
lattice . . . . . . . . . . . . . . . . . . . . . . . . . 11
atom-induced subset . . . . . 139
atomic . . . . . . . . . . . . . . . . . . . . 12
coatomic . . . . . . . . . . . . . . . . . . 12
complemented . . . . . . . . . . . . 141
empty k-pyramid . . . . . . . . . 143
empty pyramid . . . . . . . . . . . 143
212
Index
apex . . . . . . . . . . . . . . . . . . . 143
base . . . . . . . . . . . . . . . . . . . . 143
induced subset . . . . . . . . . . . 139
k-kernel . . . . . . . . . . . . . . . . . . 139
product . . . . . . . . . . . . . . . . . . 137
product with a finite set . . 137
pyramid . . . . . . . . . . . . . . . . . . 137
pyramid apex . . . . . . . . . . . . 137
pyramid base . . . . . . . . . . . . . 137
relatively complemented . . 141
lie opposite a vertex . . . . . . . . . . 164
linear . . . . . . . . . . . . . . . . . . . . . . . . 123
link
face structure . . . . . . . . . . . . . 60
versus vertex figure . . . . . . . . 20
linkages
and connectivity . . . 75, 77, 81
in polytopes . . . . . . . . . . . . . . . vii
locality of . . . . . . . . . . . . . . . . . 84
Lockeberg’s examples . . . . . . . . . viii
Lower Bound Theorem
for general polytopes . . . . . . 97
for simplicial polytopes . . . 158
lower bound theorems . . . . . . . . . 55
lower intervals . . . . . . . . . . . . . . . . . 11
ℓ-simplicial . . . . . . . . . . . . . . . . . . . . 16
Menger’s theorem . . . . . . . . . . . . . . 9
minimal linkedness . . . . . . . . 77, 81
3-polytopes. . . . . . . . . . . . . . . .80
combinatorial types . . . . 77, 93
few vertex case . . . . . . . . 84, 85
lower bound . . . . . . . viii, 76, 81
simplicial
polytopes . . . . . . 76, 77, 78
upper bound . . . . . . . . . . . . . . 83
upper bound by Gallivan . . 88
minimal positive k-spanning
configuration . . . . 169, 171
Marcus’ conjecture . . . . . . . 173
minimal positive k-spanning
set . . . . . . . . . . . . . . . . . . . 186
minimal positive spanning configuration . . . . . . . . . . . . . . . 171
minimally k-edge-connected . . 169
minor . . . . . . . . . . . . . . . . . . . . . . . . . 81
minor of a graph . . . . . . . . . . . . . . 10
missing edges . . . . . . . . . . . . . . . . 126
bound is tight . . . . . . . . . . . . 113
bound on the number . . . . 134
more on oriented matroids . . . 182
more pyramidally perfect lattices
than graded relatively complemented ones . . . . . . . 142
main theorem for polytopes . . . 13
Mani polytope . . . . . . . . . . . . . . . 158
Mani’s problem on illuminated
polytopes . . . . vii, 157, 163
Mani’s theorem on illuminated
polytopes . . . . . . . . . . . . 159
Marcus’ conjecture . . vii, 169, 173
lost counterexample . . . . . . 170
Marcus’ partial results . . . 173
on unneighborly
polytopes . . . . . . . . . . . . 173
matroid polytope . . . . . . . . 182, 186
meet-semilattice . . . . . . . . . . . . . . . 11
atom . . . . . . . . . . . . . . . . . . . . . . 12
necessarily flat . . . . . . . . . . . . . . . 109
neighborly . . . . . . . . . . . . . . . . . . . 186
neighbors in a graph. . . . . . . . . . . .8
Newman’s theorem . . . . . . . . . . . 125
no “missing edges” . . . . . . . . . . . 142
nonpolytopal PL spheres . . . . . 121
nonsimplicial Mani
polytopes . . . . . . . . 163, 164
nonsimplicial Mani polytopes
some examples . . . . . . . . . . . 166
normal . . . . . . . . . . . . . . . . . . . . . . . . 57
graph complex . . . . . . . . . . . . 60
213
Index
graph manifold . . . . . . . . . . . . 59
normalization procedure . . . . . . . 55
notation for subpaths . . . . . . . . . 78
number of combinatorial types of
k-skeleta . . . . . . . . . . . . . 106
number of empty pyramids . . . 105
number of empty simplices in simplicial polytopes . . . . . . 119
polar polytope . . . . . . . . . . . . . . . . 15
polarity . . . . . . . . . . . . . . . . . . . . . . . 15
polytopal. . . . . . . . . . . . . . . . . . . . . .48
polytopal complex . . . . . . . . . . . . . 19
face . . . . . . . . . . . . . . . . . . . . . . . 19
subcomplexes. . . . . . . . .19 – 20
polytopal face lattices . . . . . . . . 137
polytopality . . . . . . . . . . . . . . . . . . . 48
of incidence graphs . . . . . . . viii
polytope . . . . . . . . . . . . . . . . . . . . . . 13
dimension . . . . . . . . . . . . . . . . . 14
edge . . . . . . . . . . . . . . . . . . . . . . .14
face . . . . . . . . . . . . . . . . . . . . . . . 14
facet . . . . . . . . . . . . . . . . . . . . . . 14
flag . . . . . . . . . . . . . . . . . . . . . . . 14
full-dimensional . . . . . . . . . . . 14
oriented matroid of . . . . . . . 184
proper face . . . . . . . . . . . . . . . . 14
ridge . . . . . . . . . . . . . . . . . . . . . . 14
trivial face . . . . . . . . . . . . . . . . 14
truncatable . . . . . . . . . . . . . . . . 49
vertex . . . . . . . . . . . . . . . . . . . . . 14
polytopes on “few vertices”. . . .84
poset . . . . . . . . . . . . . . . . . . . . . . . . . . 10
chain . . . . . . . . . . . . . . . . . . . . . . 11
length . . . . . . . . . . . . . . . . . . . 11
cover . . . . . . . . . . . . . . . . . . . . . . 11
graded . . . . . . . . . . . . . . . . . . . . 11
has a 1̂ . . . . . . . . . . . . . . . . . . . . 11
has a 0̂ . . . . . . . . . . . . . . . . . . . . 11
interval. . . . . . . . . . . . . . . . . . . .11
isomorphism . . . . . . . . . . . . . . 11
join . . . . . . . . . . . . . . . . . . . . . . . 11
k-skeleton . . . . . . . . . . . . . . . . 139
meet . . . . . . . . . . . . . . . . . . . . . . 11
partial order. . . . . . . . . . . . . . .10
poset of structures . . . . . . . . . . . 192
positive
k-spanning set . . . . . . . . . . . 185
spanning configuration . . . 171
octahedron . . . . . . . . . . . . . . . . . . . . 14
oriented matroid
acyclic . . . . . . . . . . . . . . 181, 185
circuit system . . . . . . . . . . . . 184
convex hull . . . . . . . . . . . . . . . 185
covector lattice . . . . . . . . . . . 184
dual. . . . . . . . . . . . . . . . . . . . . .184
duality . . . . . . . . . . . . . . . . . . . 181
Las Vergnas face lattice . . 184
positive spanning set . . . . . 185
realizable. . . . . . . . . . . . . . . . .182
totally cyclic . . . . . . . . 181, 185
partially ordered set . . . see poset
path . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
connects . . . . . . . . . . . . . . . . . . . . 8
joins. . . . . . . . . . . . . . . . . . . . . . . .8
γ
Pd . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Perles’ Skeleton Theorem . . . . . vii
equivalence of different versions
of . . . . . . . . . . . . . . 105 – 107
for simplicial polytopes . . . 119
generalization to gj . . . . . . . 119
overview on bounds. . . . . . . . .5
version I. . . . . . . . . . . . .105, 114
version II . . . . . . . . . . . . . . . . 106
version III. . . . . . . . . . . . . . . .107
piecewise linear . . . . . . . . . . . . . . 123
PL ball . . . . . . . . . . . . . . . . . . . . . . 123
PL homeomorphic . . . . . . . . . . . 123
PL sphere . . . . . . . . . . . . . . . . . . . 123
Poincaré homology 3-sphere . . . 55
214
Index
positive basis . . . . . . . . . . . . . . . . see
minimal positive spanning
configuration
positive spanning set . . . . . . . . . 185
positively
k-spanning . . . . . . . . . . . . . . . 171
positively spanning . . . . . . . . . . . . 17
prism over a polytope . . . . . . . . . 18
problem by Bienia &
Las Vergnas . . . . . . vii, 181
product of polytopal lattices . 137
product of two polytopes . . . . . . 17
projectively unique . . . . . . . . . . . . 95
proper subset of a sphere . . . . . 126
pseudo-graph manifold . . . . . . . . 57
pseudo-manifold property . . . . . 56
pyramid . . . . . . . . . . . . . . . . . . . . . . . 18
k-fold . . . . . . . . . . . . . . . . . . . . . 18
pyramidal . . . . . . . . . . . . . . . . 18, 136
pyramidally inequivalent
complexes . . . . . . . . . . . . 116
of polytopes on d + 2 vertices
116
of polytopes on d + 3 vertices
116
pyramidally k-equivalent . . . . . 105
pyramidally perfect . . . . . . . . . . 136
lattices . . . . . . . . . . . . . . . . . . . 136
list of types on at most 8 elements . . . . . . . . . . . . . . . 137
versus relatively
complemented . . . . . . . . 141
rank
of a poset . . . . . . . . . . . . . . . . . 11
of an element . . . . . . . . . . . . . . 11
realizability dimension
of kernels. . . . . . . . . . . . .107
reconstruction of k-skeleton . . 104
reconstruction of skeleta . 139, 140
refinement . . . . . . . . . . . . . . . . . . . . 10
reflexivity . . . . . . . . . . . . . . . . . . . . . 10
regular cell compex
pure . . . . . . . . . . . . . . . . . . . . . . .54
regular cell complex . . . . . . . . . . . 54
antistar . . . . . . . . . . . . . . . . . . 130
cells . . . . . . . . . . . . . . . . . . . . . . . 54
Cohen-Macaulay . . . . . . . . . . 54
cone . . . . . . . . . . . . . . . . . . . . . 124
dimension . . . . . . . . . . . . . . . . . 54
face poset of. . . . . . . . . . . . . . .54
faces . . . . . . . . . . . . . . . . . . . . . . 54
facets . . . . . . . . . . . . . . . . . . . . . 54
intersection property . . . . . . 54
link . . . . . . . . . . . . . . . . . . . . . . 130
prism . . . . . . . . . . . . . . . . . . . . 124
pyramid . . . . . . . . . . . . . . . . . . 124
ridges . . . . . . . . . . . . . . . . . . . . . 54
star . . . . . . . . . . . . . . . . . . . . . . 130
strongly connected . . . . . . . . 54
strongly regular . . . . . . . . . . . 54
regular cell decomposition . . . . . 54
relation of face lattices to pyramidally perfect lattices . . 121
relative interior points . . . . . . . . . 14
relatively complemented . 12, 141,
182
Björner’s criterion . . . . . . . . . 12
versus pyramidally
perfect . . . . . . . . . . . . . . . 141
versus diamond property . . 12
remarks by McMullen . . 166 – 167
rigidity . . . . . . . . . . . . . . . . . . . . . . . . . 1
quadrilateral . . . . . . . . . . . . . . . . . . 14
question by Larman & Mani on
linkages in polytopes . . 75
low dimensions . . . . . . . 75 – 76
negative answer . . . . . . . . . . . 76
quotient
of an empty face . . . . . . . . . 126
215
Index
S γd . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
semimodular . . . . . . . . . . . . . . . . . 122
sign vector . . . . . . . . . . . . . . . . . . . 184
simple geometric statement . . . . 70
simple polytopes . . . . . . . . . . . . . . 16
simplicial polytopes . . . . . . . . . . . 16
size of minimal positive spanning
configurations . . . . . . . . 172
Gale’s Lemma . . . . . . . . . . . . 172
lower bound . . . . . . . . . . . . . . 172
upper bounds . . . . . . . . . . . . 172
standard d-crosspolytope . . . . . . 14
standard d-cube . . . . . . . . . . . . . . . 14
standard d-simplex . . . . . . . . . . . . 14
star
face structure . . . . . . . . . . . . . 60
(s, t)-linkage . . . . . . . . . . . . . . . . . . 77
strong PL d-sphere . . . . . . . . . . 123
strong chain . . . . . . . . . . . . . . . . . . . 56
strong components . . . . . . . . . . . . 58
strong walk . . . . . . . . . . . . . . . . . . . 61
concatenation . . . . . . . . . . . . . 62
construction of . . . . . . . . . . . . 63
induced components . . 61 – 62
subcomplex . . . . . . . . . . . . . . . . . . . 20
subdirect sum . . . . . . . . . . . . . . . . . 18
combinatorially . . . . . . . 18 – 19
geometrically . . . . . . . . . . . . . . 18
subdivision . . . . . . . . . . . . . . . . . . . . 10
subgraph . . . . . . . . . . . . . . . . . . . . . . . 8
subpath
notation . . . . . . . . . . . . . . . . . . . 78
sunflower. . . . . . . . . . . . . . . . . . . . .115
core . . . . . . . . . . . . . . . . . . . . . . 115
petals . . . . . . . . . . . . . . . . . . . . 115
topological representation
theorem . . . . . . . . . 182, 186
totally cyclic . . . . . . . . . . . . . . . . . 185
transitivity . . . . . . . . . . . . . . . . . . . . 10
truncatable. . . . . . . . . . . . . . . . . . . .49
underlying space . . . . . . . . . . . . . . 20
of a geometric simplicial complex . . . . . . . . . . . . . . . . . . 123
unique Mani polytopes . . . . . . . 163
unneighborly polytope . . . . . . . 171
Marcus’ conjecture . . . . . . . 170
unneighborly polytopes . . . . . . . . 97
upper bound on minimal positive
k-spanning sets . . . . . . . 187
upper intervals . . . . . . . . . . . . . . . . 11
values of the function k(d). . . . .96
vector configuration . . . . . . . . . . . 17
vertex figure . . . . . . . . . . . . . . . . . . 15
vertex-induced . . . . . . . . . . 103, 125
vertices. . . . . . . . . . . . . . . . . . . . . . . . .7
weak d-graph manifold . . . . . . . . 56
weak graph manifold . . . . . . . . . . 56
weakly normal . . . . . . . . . . . . . . . . 57
graph complex . . . . . . . . . . . . 60
graph manifold . . . . . . . . . . . . 59
theory of illumination of convex
bodies . . . . . . . . . . . . . . . . 155
theory of minors . . . . . . . . . . . . . . 75
topological minor . . . . . . . . . . 10, 81
216

Incidence Graphs and Unneighborly Polytopes - DepositOnce

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