from Wac law Marzantowicz at to Univerza na

Transcription

Introduction
Main theorems
References
Bourgin-Yang theorems for the p-toral groups
from Waclaw Marzantowicz at
to Univerza na Primorskem
Thursday 09.06.2016; under the spirit of
Bourgin-Yang theorems for the p-toral
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Applications of B-U to Combinatorics
Borsuk-Ulam versus Bourgin-Yang
In 1933 S. Ulam posed and K. Borsuk (Charles Badger) showed
that if n > m
then it is impossible to map f : S n → S m
preserving symmetry: f (−x) = −f (x) .
Next in 1954-55, C. T. Yang, and D. Bourgin, showed that if
f : Sn → Rm+1 preserves this symmetry then
dim f −1 (0) ≥ n − m − 1.
We will present versions of the latter for some other groups of
symmetries and also discuss the case n = ∞.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Abstract
Let V and W be orthogonal representations of a compact Lie
group G with V G = W G = {0}.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Abstract
Let S(V ) be the sphere of V and f : S(V ) → W be a
G -equivariant mapping.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Abstract
We estimate the dimension of set Zf = f −1 {0} in terms of
dim V and dim W , if G is the torus Tk , the p-torus Zkp , or the
cyclic group Zpk , p-prime.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Abstract
We estimate the dimension of set Zf = f −1 {0} in terms of
dim V and dim W , if G is the torus Tk , the p-torus Zkp , or the
cyclic group Zpk , p-prime.
Finally, we show that for any p-toral group:
e ,→ Tk ,→ G → P → e, P a finite p-group,
and a G -map f : S(V ) → W , with dim V = ∞ and
dim W < ∞, we have dim Zf = ∞.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Proof of the Borsuk-Ulam
Z
Suppose that f : S(Rn+1 ) →2 S(Rm+1 ) and n > m is such a map.
Let ι : Rm+1
Rn+1 natural embedding.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Z
Let ι : Rm+1
Then:
i) deg (ι ◦ f ) = 0, since if factorizes through S m
S n;
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Z
Let ι : Rm+1
Then:
S n;
ii) deg(ι ◦ f ) ∼
= 1 mod (2), more difficult !.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Z
Let ι : Rm+1
Then:
S n;
ii) deg(ι ◦ f ) ∼
= 1 mod (2), more difficult !.
i) and ii) lead to a contradiction.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Applications
Theorem (Consequences of B-U theorem)
a) No subset of Rn is homeomorphic to S n .
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Applications
b) The Lusternik-Schnirelmann theorem: If the sphere S n is
covered by n + 1 open sets, then one of these sets contains a
pair (x, −x) of antipodal points.
This is equivalent to the Borsuk-Ulam theorem
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Applications
c) The Ham sandwich theorem: For any compact sets
A1 , . . . , An in Rn we can always find a hyperplane dividing
each of them into two subsets of equal measure.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Applications
c) The Ham sandwich theorem: For any compact sets
A1 , . . . , An in Rn we can always find a hyperplane dividing
each of them into two subsets of equal measure.
d) The Brouwer fixed-point theorem.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Example
d) Common practical applications: The case n = 2:
∃ (x, −x) on the Earth’s surface s.t. T (x) = T (−x),
P(x) = P(−x)
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Example
d) Common practical applications: The case n = 2:
∃ (x, −x) on the Earth’s surface s.t. T (x) = T (−x),
P(x) = P(−x)
e) The case n = 1: there always ∃ a pair of opposite points on
the earth’s equator with the same temperature.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Necklace splitting problem
Suppose a necklace, open at the clasp, has k · n beads,
k · ai beads of colour i, 1 ≤ i ≤ t.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Then the necklace splitting problem: find a partition of the
necklace into k parts (not necessarily contiguous),
each with exactly ai beads of colour i; called a k-split.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
If colours are contiguous then any k splitting must contain at least
k − 1 cuts, so the size is at least (k − 1)t.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
If colours are contiguous then any k splitting must contain at least
k − 1 cuts, so the size is at least (k − 1)t.
Bourgin-Yang
theorems for the p-toral
N. Alon and D.B.West used the Borsuk-Ulam
theorem:
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
The Kneser graph: KGn,k : vertices correspond to the k-element
subsets of a set of n elements,
two vertices are adjacent iff the corresponding sets are disjoint.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Martin Kneser (1955) conjecture:
the chromatic number of KGn,k is exactly n − 2k + 2
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
László Lovász (1978) proved this using topological methods.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
László Lovász (1978) proved this using topological methods.
Imre Bárány (1978) gave a simple proof, using the Borsuk-Ulam t.
and a lemma of David Gale.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Theorem (Ham sandwich theorem for measures)
Let µ1 , µ2 , ..., µd be finite Borel measures on Rd such that every
hyperplane has measure 0 for each of the µi (we refer to such
measures as mass distributions).
Then there exists a hyperplane h such that µi (h+ ) = 12 µi (Rd ) for
i = 1, 2, .., d, where h+ denotes one of the half-spaces defined by h.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Theorem (Ham sandwich theorem for measures)
Let µ1 , µ2 , ..., µd be finite Borel measures on Rd such that every
hyperplane has measure 0 for each of the µi (we refer to such
measures as mass distributions).
Then there exists a hyperplane h such that µi (h+ ) = 12 µi (Rd ) for
i = 1, 2, .., d, where h+ denotes one of the half-spaces defined by h.
Theorem (Ham sandwich theorem for point sets)
Let A1 , A2 , ..., Ad ⊂ Rd be finite point sets. Then there exists a
hyperplane h that simultaneously bisects A1 , A2 , ..., Ad .
The idea of proof is very simple: replace the points of Ai by tiny
balls and apply the ham sandwich theorem for measures. But there
are some subtleties along the way.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Borsuk-Ulam
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Borsuk-Ulam
The ham-sandwich theorem in R3 with three ingredients
The pancakes theorem in R2 with two ingredients = each of
distinct cake.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Bourgin-Yang
The pancakes theorem in R3 : two pancakes ⊂ R2 ⊂ R3
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Bourgin-Yang
The pancakes theorem in R3 : two pancakes ⊂ R2 ⊂ R3
Each plane cutting R2 along the line of solution in R2 gives a
solution in R3 . Their unit normal vectors form Zf = S1 ⊂ S(R3 )
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Borsuk-Ulam: Hundreds of generalizations & several applications
See the article of Steinlein [25] for generalizations
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Fibrewise setting: f : S(E ) ⊂ E −→ E 0 over B.
1981- Jaworowski [12];
1988 - Dold [10] for G = Z2 ;
1989 - Nakaoka [23] for G = Z2 , Zp and S 1 ;
1990 - Izydorek and Rybcki [11] for G = Zp ;
1995 - Kosta-Mramor [19] for Banach vector bundles;
2007 - de Mattos and dos Santos [9] for G = Zp , and a
product of spheres;
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Fibrewise setting: f : S(E ) ⊂ E −→ E 0 over B.
1981- Jaworowski [12];
1988 - Dold [10] for G = Z2 ;
1989 - Nakaoka [23] for G = Z2 , Zp and S 1 ;
1990 - Izydorek and Rybcki [11] for G = Zp ;
1995 - Kosta-Mramor [19] for Banach vector bundles;
2007 - de Mattos and dos Santos [9] for G = Zp , and a
product of spheres;
Applications:
Nonlinear Analysis: SeeTopological Methods for Variational
Problems with Symmetries of T. Bartsch [3]
Combinatorics: See Using the Borsuk-Ulam Theorem of J.
Matousek [18]
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
”Classical” Bourgin-Yang theorem
– Bourgin-Yang theorem for the mappings of spheres
G
representation f : S(V )→W
2012 - W. M. , de Mattos and dos Santos [16]
for G = Zpk , a use of equivariant K -theory;
2013-2015 - W. M., de Mattos and dos Santos [17]
for G = (Zp )k , (Z2 )k , (S 1 )k , a use of Borel cohomology;
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
”Classical” Bourgin-Yang theorem
– Bourgin-Yang theorem for the mappings of spheres
G
representation f : S(V )→W
2012 - W. M. , de Mattos and dos Santos [16]
for G = Zpk , a use of equivariant K -theory;
2013-2015 - W. M., de Mattos and dos Santos [17]
for G = (Zp )k , (Z2 )k , (S 1 )k , a use of Borel cohomology;
– 2016 - W.M., Blaszczyk, Singh [4] for a general setting: X , Y
more general,
a combination of methods.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
The classical Bourgin-Yang problem studied here is
similar but different than the Bourgin-Yang, or correspondingly
Borsuk-Ulam problem for coincidence points along an orbit.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
The latter studied for G = Zpk by Munkholm and for G = Zkp by
Volovikov in several papers (cf. [20, 21, 22] and respectively
[26, 27, 28] with references there).
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
Called Borsuk-Ulam, or respectively Bourgin-Yang type theorem.
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
It studies
dim A(f ) = {x ∈ X : | f (x) = f (gx), for all g ∈ G }
for a map (not equivariant in general) f : X → Y of G -spaces X
and Y .
Introduction
Main theorems
References
Roots
Abstract
Borsuk-Ulam theorem
It studies
dim A(f ) = {x ∈ X : | f (x) = f (gx), for all g ∈ G }
for a map (not equivariant in general) f : X → Y of G -spaces X
and Y .
There are relations but not direct - we will not discuss the latter.
Introduction
Main theorems
References
Classical proof of Bourgin-Yang theorem
The length in equivariant cohomology theory
Bourgin-Yang theorem for G = (Zp )k and G = (S1 )k
Generalization
Characterization of p-toral groups
Bourgin-Yang for the cyclic group G = Zp k
Idea od proof
Theorem (Yang, Bourgin)
Z
2
If f : S(Rn+1 ) −→
Rm+1 then dim Zf ≥ n − m − 1.
An invariant of free Z2 -space. Put G = Z2
Introduction
Main theorems
References
Generalization
Idea od proof
Z
2
If f : S(Rn+1 ) −→
X a free G -space (metric, CW-complex),
φ : X /G → BG = RP(∞) a map classifying p : X → X /G .
γ ∈ H 1 (BG ; F2 ) , here H ∗ (BG ; F2 ) = F2 [γ].
Definition:
iG (X ) = smallest k : φ∗ (γ k ) = 0
Introduction
Main theorems
References
Generalization
Idea od proof
Z
2
If f : S(Rn+1 ) −→
Definition:
Properties:
G
iG (Y ∪ X ) ≤ iG (Y ) + iG (X ); f : X → Y =⇒ iG (X ) ≤ iG (Y );
Introduction
Main theorems
References
Generalization
Idea od proof
Z
2
If f : S(Rn+1 ) −→
Definition:
Properties:
G
G
∀ A = clA ∃ U ⊃ A , U open, s.t.
iG (U) = iG (A)
Introduction
Main theorems
References
Generalization
Idea od proof
Z
2
If f : S(Rn+1 ) −→
Definition:
Properties:
G
G
∀ A = clA ∃ U ⊃ A , U open, s.t. iG (U) = iG (A)
iG (S n , −) = n + 1; iG (XBourgin-Yang
) ≤ coh dim Xtheorems
+ 1.
for the p-toral
Introduction
Main theorems
References
Generalization
Proof of Bourgin-Yang
G
Proof: Take V = S(Rn+1 ) \ Zf , and Zf ⊂ U s.t. iG (U) = iG (Zf ).
Introduction
Main theorems
References
Generalization
G
By the first two properties of iG :
n + 1 = iG (S(Rn+1 )) ≤ iG (V ) + iG (U) ,
Introduction
Main theorems
References
Generalization
G
n + 1 = iG (S(Rn+1 )) ≤ iG (V ) + iG (U) ,
But iG (V ) ≤ iG (S(Rm+1 )) = m + 1,
because f : V → Rm+1 \ {0} ∼ S(Rm+1 ).
Introduction
Main theorems
References
Generalization
G
n + 1 = iG (S(Rn+1 )) ≤ iG (V ) + iG (U) ,
But iG (V ) ≤ iG (S(Rm+1 )) = m + 1,
because f : V → Rm+1 \ {0} ∼ S(Rm+1 ).
It gives
iG (Zf ) = iG (U) ≥ n + 1 − m + 1,
and using the last property we get
Introduction
Main theorems
References
Generalization
G
n + 1 = iG (S(Rn+1 )) ≤ iG (V ) + iG (U) ,
But iG (V ) ≤ iG (S(Rm+1 )) = m + 1,
because f : V → Rm+1 \ {0} ∼ S(Rm+1 ).
It gives
iG (Zf ) = iG (U) ≥ n + 1 − m + 1,
and using the last property we get
coh.dimZf ≥ n − m − 1 .
Introduction
Main theorems
References
Generalization
Notation
dim X : the covering dimension of X and coh.dimX the
cohomological dimension of a space X , i.e.,
coh.dimX = max{n | Ȟ n (X ) 6= 0}
where Ȟ n (−) denotes the Čech cohomology with coefficients
F = Zp or F = Q, depending on whether G = Zkp or G = Tk .
Introduction
Main theorems
References
Generalization
Notation
We have coh.dimX ≤ dim X .
Introduction
Main theorems
References
Generalization
Notation
We have coh.dimX ≤ dim X .
H ∗ (−)
Also, H∗ (−),
(H̃∗ (−), H̃ ∗ (−)) the (reduced) singular
(co)homology with coefficients F = Zp or F = Q, depending on
whether G = Zkp or G = Tk .
Introduction
Main theorems
References
Generalization
Recall that for G = Zkp , with p prime odd, and G = Tk ∀
nontrivial irreducible orthogonal representation
is even dimensional and admits the complex structure,
=⇒ V and W admit it too.
Introduction
Main theorems
References
Generalization
Denote d(V ) = dimC V = 12 dimR V , and correspondingly
d(W ) = dimC W = 12 dimR W .
Introduction
Main theorems
References
Generalization
If G = Zk2 and V ,W are orthogonal representations of G , then
denote d(V ) = dimR V , and respectively d(W ) = dimR W .
Introduction
Main theorems
References
Generalization
If G = Zk2 and V ,W are orthogonal representations of G , then
denote d(V ) = dimR V , and respectively d(W ) = dimR W .
For a G -map f : S(V ) → W we study the set
Zf := f −1 (0) ⊂ S(V )
Introduction
Main theorems
References
Generalization
Let A be a set of G -spaces, h∗ a multiplicative equivariant
cohomology theory, and I ⊆ h∗ (pt) an ideal.
Definition
The (A, h∗ , I )-length of a G -space X is defined to be the smallest
integer k ≥ 1 such that there exist A1 , . . . , Ak ∈ A with the
property that for any αi ∈ I ∩ ker h∗ (pt) → h∗ (Ai ) , 1 ≤ i ≤ k,
pX∗ (α1 ) ^ · · · ^ pX∗ (αk ) = 0,
where pX : X → pt.
Introduction
Main theorems
References
Generalization
Theorem ([3, Theorem 4.7])
The length has the following properties:
(1) If there ∃ an h∗ -functorial G -equivariant map X → Y , then
`(X ) ≤ `(Y ).
(2) Let A, B ⊆ X be G -invariant subspaces such that
^
h∗ (X , A) × h∗ (X , B) −→ h∗ (X , A ∪ B)
is defined. If A ∪ B = X , then `(X ) ≤ `(A) + `(B).
(3) If h∗ = HG∗ , I is noetherian and X is paracompact, then any
GA = A, cl A = A ⊆ X has an open G -neighborhood U ⊆ X
such that `(U) = `(A).
Introduction
Main theorems
References
Generalization
Theorem ([3, Theorem 4.7])
The length has the following properties:
(1) If there ∃ an h∗ -functorial G -equivariant map X → Y , then
`(X ) ≤ `(Y ).
(2) Let A, B ⊆ X be G -invariant subspaces such that
^
h∗ (X , A) × h∗ (X , B) −→ h∗ (X , A ∪ B)
is defined. If A ∪ B = X , then `(X ) ≤ `(A) + `(B).
(3) If h∗ = HG∗ , I is noetherian and X is paracompact, then any
GA = A, cl A = A ⊆ X has an open G -neighborhood U ⊆ X
such that `(U) = `(A).
Depending on the group G , we set:
(1) if G = (Z2 )k : h∗ = HG∗ (−; Z2 ), I = HG∗ (pt; Z2 ),
(2) if G = (Zp )k , p > 2: h∗ = HG∗ (−; Zp ), I = (c1 , . . . , ck ),
(3) if G = (S1 )k : h∗ = HG∗ (−; Q), Bourgin-Yang
I = HG∗ (pt; Q).theorems for the p-toral
Introduction
Main theorems
References
Generalization
Not difficult to repeat and get
`(Zf ) ≥ `(S(V )) − `(S(W )) = d(V ) − d(W )
Introduction
Main theorems
References
Generalization
`(Zf ) ≥ `(S(V )) − `(S(W )) = d(V ) − d(W )
For G = (Z2 )k , or correspondingly G = Zp , p > 2, and G = S1
one can show that any G -invariant closed set
`(Z ) ≤ coh.dimZ + 1 , or respectively 2`(Z ) ≤ coh.dim Z + 1 .
Introduction
Main theorems
References
Generalization
`(Zf ) ≥ `(S(V )) − `(S(W )) = d(V ) − d(W )
For G = (Z2 )k , or correspondingly G = Zp , p > 2, and G = S1
one can show that any G -invariant closed set
`(Z ) ≤ coh.dimZ + 1 , or respectively 2`(Z ) ≤ coh.dim Z + 1 .
The case G = (S1 )k can be reduced to G = S1 . However it is not
possible to compare `(Z ) and coh.dim Z for the group G = (Zp )k ,
p > 2.
Introduction
Main theorems
References
Generalization
Theorem
Let V , W be two orth. representations of G = Zkp or G = Tk such
G
that V G = W G = {0}. If f : S(V )→W is G -map, then
coh.dimZf ≥ dimR V − dimR W − 1 .
Introduction
Main theorems
References
Generalization
Theorem
G
In part., if dimR W < dimR V , then 6 ∃ G -map f : S(V ) → S(W ).
Introduction
Main theorems
References
Generalization
Theorem
G
In part., if dimR W < dimR V , then 6 ∃ G -map f : S(V ) → S(W ).
Corollary
Let G = Zkp with p > 2, or G = Tk and V , W as above. Then for
G
any f : S(V )→W dimR V > dimR W implies coh.dimZf ≥ 1 .
Indeed, dimR V − dimR W − 1 = 2d(V ) − 2d(W ) − 1
is integral, positive and odd.
Introduction
Main theorems
References
Generalization
We shall use a most general version of the Borsuk-Ulam theorem
by Assadi in [2, page 23] (for p-torus) and Clapp and Puppe in [8,
Theorem 6.4] for the torus and p-torus
Introduction
Main theorems
References
Generalization
We shall use a most general version of the Borsuk-Ulam theorem
by Assadi in [2, page 23] (for p-torus) and Clapp and Puppe in [8,
Theorem 6.4] for the torus and p-torus
Theorem
Let G be a p-torus or a torus. Let X and Y be G -spaces with
fixed-points-free actions; moreover, in the case of a torus action
assume additionally that Y has finitely many orbit types. Suppose
that H̃j (X ) = H̃ j (X ) = 0 for j < n, Y is compact or paracompact
and finite-dimensional, and Hj (Y ) = H j (Y ) = 0 for j ≥ n. Then
there exists no G -equivariant map of X into Y .
Introduction
Main theorems
References
Generalization
Proof: Denote m = dimR V and n = dimR W and suppose
coh.dimZf < m − n − 1.
Then,
Ȟ i (Zf ) = 0, for any i > m − n − 2.
Introduction
Main theorems
References
Generalization
Proof: Denote m = dimR V and n = dimR W and suppose
coh.dimZf < m − n − 1.
Then,
Ȟ i (Zf ) = 0, for any i > m − n − 2.
By using Poincaré-Alexander-Lefschetz duality and the long exact
sequence of the pair
(SV , SV \ Zf ), we conclude
0 = Ȟ i (Zf ) = Hm−1−i (SV , SV \Zf ) = H̃m−i−2 (SV \Zf ), for j = m−i−2 <
H̃j (SV \ Zf ) = 0, for j < n.
Introduction
Main theorems
References
Generalization
On the other hand, we have
Hj (W \ {0}) = Hj (SW ) = 0, for j ≥ n.
Introduction
Main theorems
References
Generalization
Hj (W \ {0}) = Hj (SW ) = 0, for j ≥ n.
However,
f : SV \ Zf → W \ {0}
is a G -equivariant map, which contradicts Theorem 4.
Introduction
Main theorems
References
Generalization
Hj (W \ {0}) = Hj (SW ) = 0, for j ≥ n.
However,
f : SV \ Zf → W \ {0}
is a G -equivariant map, which contradicts Theorem 4.
In particular, if dimR V > dimR W , for a G -map
f : S(V ) → W \ {0} ⊂ W it implies that coh.dimZf ≥ 0 and,
consequently, Zf 6= ∅, which gives a contradiction.
Introduction
Main theorems
References
Generalization
Theorem
Let G = (Z2 )k , (Zp )k or (S1 )k , with k ≥ 1.
- Let X be a G -space and a K -orientable closed topological
manifold such that H̃ i (X ) = 0 for i < n − 1.
Introduction
Main theorems
References
Generalization
Theorem
- Let Y be a G -space and A ⊂ Y a G -subspace such that
Y − A is compact (or paracompact and finite dimensional) and
H i (Y − A) = 0 for i ≥ m.
Additionally (Y − A)G = ∅.
In the case p = 0, i.e. the torus, suppose Y has finitely many
orbit type.
Introduction
Main theorems
References
Generalization
Theorem
- Let Y be a G -space and A ⊂ Y a G -subspace such that
Y − A is compact (or paracompact and finite dimensional) and
H i (Y − A) = 0 for i ≥ m.
Additionally (Y − A)G = ∅.
In the case p = 0, i.e. the torus, suppose Y has finitely many
orbit type.
If f : X −→ Y is a G -map, then
dim f −1 (A) ≥ n − m − 1.
Introduction
Main theorems
References
Generalization
Theorem (Characterization of p-toral groups, WM/DM/ES (12))
a) Let G be a p-toral group 1 ,→ Tk → G → P → 1.
Then for the sphere S(V ) of a G -Hilbert space (orthogonal
representation) V , V G = {0}, dim V = ∞ , and finite
dimensional orthogonal representation W of G such that
W G = {0}, and a G -map f : S(V ) → W we have
dim Zf = l(Zf ) = ∞ .
Introduction
Main theorems
References
Generalization
Theorem (Characterization of p-toral groups, WM/DM/ES (12))
a) Let G be a p-toral group 1 ,→ Tk → G → P → 1.
Then for the sphere S(V ) of a G -Hilbert space (orthogonal
representation) V , V G = {0}, dim V = ∞ , and finite
dimensional orthogonal representation W of G such that
W G = {0}, and a G -map f : S(V ) → W we have
dim Zf = l(Zf ) = ∞ .
b) If G is not p-toral then ∃ an infinite-dim. fixed point free
G -Hilbert space V , a finite dimensional representation W of
G with W G = {0} and a G -map f : S(V ) → W such that
Zf = ∅, e.g. dim Zf = −1 < ∞ .
Introduction
Main theorems
References
Generalization
An adaptation of proof of a B-U theorem of Bartsch, Clapp & D.
Puppe, based on the Borel cohomology of stable cohomotopy
theory and used the Segal conjecture (G. Carlson)
[) = π 0 (BG ).
A(G
st
Introduction
Main theorems
References
Generalization
[) = π 0 (BG ).
A(G
st
0 (X × EG ) the
i) Equivariant cohomology theory hG∗ (X ) = ωst
G
Borel construction for the stable cohomotopy theory;
Introduction
Main theorems
References
Generalization
[) = π 0 (BG ).
A(G
st
0 (X × EG ) the
G
ii) By the mentioned theorem
0 (BG ) = π 0 (BG ) = A(G
[),
hG∗ (pt) = hG0 (pt) = ωst
st
where the completion is taken with respect to the ideal
I := ker dim : A(G ) → Z.
Introduction
Main theorems
References
Generalization
[) = π 0 (BG ).
A(G
st
0 (X × EG ) the
G
ii) By the mentioned theorem
0 (BG ) = π 0 (BG ) = A(G
[),
hG∗ (pt) = hG0 (pt) = ωst
st
where the completion is taken with respect to the ideal
I := ker dim : A(G ) → Z.
iii) Next, one should use the fact that the completion map
[) is injective if P is a finite p-group (E. Laitinen).
A(G ) → A(G
Introduction
Main theorems
References
Generalization
Let `(X ) be the G -length with respect the above hG∗ (X )
Introduction
Main theorems
References
Generalization
Take X = S(V ), V infinite-dimensional Hilbert space. Known [3]
that `(X ) = `(pt) if X is contractible G -space.
Introduction
Main theorems
References
Generalization
On the other hand for any finite G there exists an element
α ∈ A(G ) such that αn 6= 0 for every n ∈ N (T. tom Dieck).
[) = ω 0 (BG ) = h0 (pt) has the
Consequently, its image α
b ∈ A(G
st
G
same property for a p-group. This shows that `(pt) = ∞.
Introduction
Main theorems
References
Generalization
On the other hand for any finite G there exists an element
α ∈ A(G ) such that αn 6= 0 for every n ∈ N (T. tom Dieck).
[) = ω 0 (BG ) = h0 (pt) has the
Consequently, its image α
b ∈ A(G
st
G
same property for a p-group. This shows that `(pt) = ∞.
From it follows that for every G -map f : S(V ) → W
`(Zf ) ≥ `(S(V )) − `(S(W )) = ∞
as `(S(W )) < ∞. The remaining task is to adapt it for any toral
p-group and show that also dim Zf = ∞.
Introduction
Main theorems
References
Generalization
Bourgin-Yang for the cyclic group Zpk , k ≥ 2
V an orthogonal repr. of G = Zpk such that V G = {0}.
Introduction
Main theorems
References
Generalization
(
dimR if p = 2
Notation: d(V ) =
dimC = 12 dimR if p is odd
Introduction
Main theorems
References
Generalization
(
dimR if p = 2
Notation: d(V ) =
For given two powers 1 ≤ m ≤ n ≤ p k−1
let Am,n := {G /H | H ⊂ G ; m ≤ |H| ≤ n },
(1)
Introduction
Main theorems
References
Generalization
(
dimR if p = 2
Notation: d(V ) =
let Am,n := {G /H | H ⊂ G ; m ≤ |H| ≤ n },
(1)
Theorem (D. de Mattos, E. dos Santos, WM (12))
Let V , W two orthogonal representations of G = Zpk , p prime,
G
k ≥ 1, such that V G = W G = {0}. Let f : S(V ) → W .
If AS(V ) ⊂ Am,n and AS(W ) ⊂ Am,n then
Introduction
Main theorems
References
Generalization
(
dimR if p = 2
Notation: d(V ) =
let Am,n := {G /H | H ⊂ G ; m ≤ |H| ≤ n },
(1)
Theorem (D. de Mattos, E. dos Santos, WM (12))
Let V , W two orthogonal representations of G = Zpk , p prime,
G
k ≥ 1, such that V G = W G = {0}. Let f : S(V ) → W .
If AS(V ) ⊂ Am,n and AS(W ) ⊂ Am,n then
dim(Zf ) ≥ 2
h (d(V ) − 1)m i
Bourgin-Yang
theorems
− d(W ) := φ(V
, W ). for the p-toral
Introduction
Main theorems
References
Generalization
Corollaries
Corollary (already from Bartsch B–U for G = Zpk )
In particular, if d(W ) < d(V )/p k−1 , then dim Zf ≥ 0,
=⇒ no G -equivariant map f : S(V ) → S(W ).
Introduction
Main theorems
References
Generalization
Corollaries
Example given by T. Bartsch
If 1 ,→ G0 ,→ G → Γ → 1 is a compact Lie group
and Γ has an element of order p 2 , then there exist V ,W
orthogonal repr. V G = {0} = W G , dim W < dim V
and a G -equivariant map f : S(V ) → S(W ).
Introduction
Main theorems
References
Generalization
Corollaries
Example given by T. Bartsch
If 1 ,→ G0 ,→ G → Γ → 1 is a compact Lie group
and Γ has an element of order p 2 , then there exist V ,W
orthogonal repr. V G = {0} = W G , dim W < dim V
and a G -equivariant map f : S(V ) → S(W ).
In particular if G = Γ a finite group with an element of order p 2 .
Introduction
Main theorems
References
Generalization
Tools
An index of type iZ2 for a Zpk spaces but defined by use of the
equivariant KG∗ theory.
Its definition, thus the value, depends on the orbits in X .
ln (X ) = (Am,n , KG∗ , R) − length index of (X ) .
(2)
We use the family Am,n but the invariant does not depend on n.
Introduction
Main theorems
References
Generalization
Tools
(2)
Theorem (T. Bartsch (90) - fine computation)
Let V be an orthogonal repr. of G = Zpk with V G = {0}. Fix
m, n two powers of p as above. Then
Introduction
Main theorems
References
Generalization
Tools
(2)

i
h
1 + (d−1)m if AS(V ) ⊂ Am,n ,
n
ln (S(V )) ≥
∞
if AS(V ) * A1,n ,
Introduction
Main theorems
References
Generalization
Tools
(2)

i
h
1 + (d−1)m if AS(V ) ⊂ Am,n ,
n
ln (S(V )) ≥
∞
if AS(V ) * A1,n ,
Introduction
Main theorems
References
References
C. Allday e V. Puppe, Cohomological Methods in
Transformation Groups. Cambridge Studies in Advanced
Mathematics 32, Cambridge University Press, (1993).
A. H. Assadi, Varieties in finite transformation groups. Bull.
Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 459-463.
T. Bartsch, Topological Methods for Variational Problems with
Symmetries. Lecture Notes in Mathematics 1560,
Springer-Verlag Berlin Heidelberg (1993).
Blaszczyk, Z., Marzantowicz, W. e Singh, M., General
Bourgin-Yang theorems. arxiv:1512.02399 (2015).
A. Borel, Seminar on transformation Groups. Princeton:
Princeton University Press (1960).
Introduction
Main theorems
References
K. Borsuk. Drei Sätze Über die n-dimensionale euklidische
Sphäre, Fund. Math. 20 177-190, (1933).
D. G. Bourgin, On some separation and mapping theorems.
Comment. Math. Helv. 29 (1955), 199-214.
M. Clapp e D. Puppe, Critical point theory with symmetries.
J. reine angew. Math. 418 (1991), 1-29.
D. de Mattos and E. L. dos Santos, A parametrized
Borsuk-Ulam theorem for a product of spheres with free
Zp -action and free S 1 -action, Algebraic and Geometric
Topology 7 (2007) 1791-1804.
A. Dold, Parametrized Borsuk-Ulam theorems. Comment.
Math. Helv. 63 (1988) n. 2, 275-285.
Introduction
Main theorems
References
M. Izydorek e S. Rybicki, On parametrized Borsuk-Ulam
theorem for free Zp -action. Algebraic topology (San Feliu de
Guixols 1990) 227-234, Lecture Notes in Math., 1509,
Springer, Berlin, (1992).
J. Jaworowski, A continuous version of the Borsuk-Ulam
theorem. Proc. Amer. Math. Soc., 82(1):112-114, (1981).
J. Jaworowski, Fibre-preserving maps of sphere-bundles into
vector space bundles. In Fixed point theory (Sherbrooke, Que.,
1980), Lecture Notes in Math. 886, pages 154-162. Springer,
(1981).
J. Jaworowski, Bundles with periodic maps and mod p Chern
polynomial. Proc. of the Amer. Math. Soc. 132, no. 4, (2004)
1223-1228.
Introduction
Main theorems
References
W. Marzatowicz e M. Izydorek, Equivariant maps between
cohomology spheres. Topol. Methods Nonlinear Anal. 5
(1995), 279-290.
W. Marzantowicz, D. de Mattos and E. L. dos Santos,
Bourgin-Yang version of the Borsuk-Ulam theorem for Zpk
-equivariant maps. Algebraic and Geometric Topology 12
(2012) 2146-2158.
W. Marzantowicz, D. de Mattos e E. L. dos Santos,
Bourgin-Yang versions of the Borsuk-Ulam theorem for p-toral
groups. arXiv:1302:1494 (2014).
J. Matousek, Using the Borsuk-Ulam Theorem. Universitext,
Springer-Verlag,(2002).
Introduction
Main theorems
References
N. Mramor-Kosta, A parametrized Borsuk-Ulam theorem for
Banach bundles. Glas. Mat. Ser. III 30(50) (1995), no. 1,
111-127.
H. J. Munkholm, A Borsuk-Ulam theorem for maps from a
sphere to a compact topological manifold. Illinois J. Math. 13
(1969), 116-124.
H. J. Munkholm, Borsuk-Ulam type theorems for proper Zp
-actions on ( mod p homology) n- spheres, Math. Scand. 24
(1969), 167-185.
H. J. Munkholm, On the Borsuk-Ulam theorem for Zpk -actions
on S 2n−1 and maps S 2n−1 −→ Rn . Math. Scand. 24 (1969),
167-185.
M. Nakaoka, Parametrized Borsuk-Ulam theorems and
characteristic polynomials. Topological fixed point theory and
Introduction
Main theorems
References
applications (Tianjin, 1988), 155-170. Lecture Notes in Math.
1411, Springer, Berlin, (1989).
G. Segal, Equivariant K -theory. Publ. Math. IHES 34 (1968)
129-151.
H. Steinlein, Borsuk’s antipodal theorem and its
generalizations and applications: a survey. Topological
methods in nonlinear analysis, 166-235, Sm. Math. Sup. 95,
Presses Univ. Montreal, Montreal, QC, (1985).
A. Volovikov, The Bourgin-Yang theorem, Uspekhi Mat. Nauk
35 (1980), 159-162 (Russian).
A. Volovikov, A theorem of Bourgin-Yang type for
(Zp )n -actions, Mat. Sb. 183 (1992), 115-144. (Russian;
English translation in Russia Acad. Sci. Sb.
Introduction
Main theorems
References
A. Y. Volovikov, On a topological generalization of Tveberg’s
theorem, (Russian) Mat. Zametki 59 (1996), n. 3, 454-456;
translation in Math. Notes 59 (1996), n. 3-4, 324-325.
C. T. Yang, On the theorems of Borsuk-Ulam,
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C. T. Yang, On Theorems of Borsuk-Ulam,
Kakutani-Yamabe-Yujobo and Dyson, II. Annals of
Mathematics, Second Series, Vol. 62, No. 2 (Sep., 1955), pp.
271-283.

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