the methods of algebraic topology from the viewpoint of cobordism

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THE METHODS OF ALGEBRAIC TOPOLOGY FROM THE VIEWPOINT OF COBORDISM
THEORY
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1967 Math. USSR Izv. 1 827
(http://iopscience.iop.org/0025-5726/1/4/A06)
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Izv. Akad. Nauk SSSR
Ser. Mat. Tom 31 (1967), No. 4
Math. USSR - Izvestija
Vol. 1 (1967), No. 4
THE METHODS OF ALGEBRAIC TOPOLOGY FROM
THE VIEWPOINT OF COBORDISM THEORY*
S. P. NOVIKOV
UDC 513-83
The goal of this work is the construction of the analogue to the Adams spectral sequence
in cobordism theory, calculation of the ring of cohomology operations in this theory, and
also a number of applications: to the problem of computing homotopy groups and the classical Adams spectral sequence, fixed points of transformations of period p, and others.
Introduction
In algebraic topology during the last few years the role of the so-called extraordinary homology
and cohomology theories has started to become apparent; these theories satisfy all the EilenbergSteenrod axioms, except the axiom on the homology of a point. The merit of introducing such theories
into topology and their first brilliant applications are due to Atiyah, Hirzebruch, Conner and Floyd,
although in algebraic geometry the germs of such notions have appeared earlier (the Chow ring, the
Grothendieck K-functor, etc.). Duality laws of Poincare type, Thorn isomorphisms, the construction
of several important analogues of cohomology operations and characteristic classes, and also relations
between different theories were quickly discovered and understood (cf. [2> 4, 5, 8, 9, 11, 12]).
These ideas and notions gave rise to a series of brilliant results ( L 2-13-1). In time there became
manifest two important types of such theories: (1) theories of "K type" and (2) theories of "cobordism
type" and their dual homology ("bordism") theories.
The present work is connected mainly with the theory of unitary cobordism. It is a detailed account and further development of the author's work [19], The structure of the homology of a point in
the unitary cobordism theory was first discovered by Milnor [ 15] and the author [ I 7 ] ; the most complete and systematic account together with the structure of the ring can be found in [ I 8 ] . Moreover,
in recent work of Stong L^2J and Hattori important relations of unitary cobordism to ϋ-theory were
found. We freely use the results and methods of all these works later, and we refer the reader to the
works L 1' > 17, 18, 22j for preliminary information.
Our basic aim is the development of new methods which allow us to compute stable homotopy invariants in a regular fashion with the help of extraordinary homology theories, by analogy with the
method of Cartan-Serre-Adams in the usual classical Zp-cohomology theory. We have succeeded in
*Editor's note: Some small additions, contained in braces j j , have been made in translation.
827
828
S. P. NOVIKOV
the complete computation of the analogue of the Steenrod algebra and the construction of a "spectral
sequence of Adams type"* in some cohomology theories, of which the most important is the theory of
i/-cobordism, and we shall sketch some computations which permit us to obtain and comprehend from
the same point of view a series of already known concrete results (Milnor, Kervaire, Adams, ConnerFloyd, and others), and some new results as well.
In the process of the work the author ran into a whole series of new and tempting algebraic and
topological situations, analogues to which in the classical case are either completely lacking or
strongly degenerate; many of them have not been considered in depth. All this leads us to express
hope for the perspective of this circle of ideas and methods both for applications to known classical
problems of homotopy theory, and for the formulation and solution of new problems from which one can
expect the appearance of nontraditional algebraic connections and concepts.
The reader, naturally, is interested in the following question: to what extent is the program (of
developing far-reaching algebraic-topological methods in extraordinary cohomology theory) able to
resolve difficulties connected with the stable homotopy groups of spheres? In the author's opinion, it
succeeds in showing some principal (and new) sides of this problem, which allow us to put forth arguments about the nearness of the problems to solution and the formulation of final answers. First of
all, the question should be separated into two parts: (1) the correct selection of the theory of cobordism type as "leading" in this program, and why it is richer than cohomology and K-theory; (2) how
to look at the problem of homotopy groups of spheres from the point of view of cobordism theory.
The answer to the first part of the question is not complicated. As is shown in Appendix 3, if
we have any other "good" cohomology theory, then it has the form of cobordism with coefficients in
an Ω-module. Besides, working as in § § 9 and 12, it is possible to convince oneself that these give
the best filtrations for homotopy groups (at any rate, for complexes without torsion; for ρ = 2 it may
be that the appropriate substitute for MU is MSU). In this way, the other theories lead to the scheme
of cobordism theory, and there their properties may be exploited in our program by means of homological algebra, as shown in many parts of the present work.
We now attempt to answer the .second fundamental part of the question. Here we must initially
formulate some notions and assertions. Let AU lAU j be the ring of cohomology operations in U*theory [U*, respectively], Λ ρ = U*p(P), A = U*(P), Ρ = point, Qp = p-adic integers.** Note that
A c F . The ring over Qp, Λ ® ZQ p D Λ ρ , lies in A V <g) z Qp D A V, and Λ ® ζ Qp is a local ring
u
with maximal ideal m C Λ ® z Qp, where h ® z Qp/m = Ζp. Note that Λ ρ is an A ^-module and A is
also a left Ap-module.
Consider the following rings:
mv = m Π Λ ρ ,
Λρ / mp =
Zp,
*It may be shown that the Adams spectral sequence is the generalization specifically for S-categories (see
§ 1) of "the universal coefficient formula," and this is used in the proofs of Theorems 1 and 2 of Appendix 3**f/*-theory is a direct summand of the cohomology theory U* ® Qp, having spectrum Mp such that
H* (Mp, Z p ) = Α/(βΑ + Α β) j where A is the Steenrod algebra over Zp and β is the Boksteih operator! (see
§§ 1, 5, 11, 12).
METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY
A = APU = 2
829
τηρ*Αρυ/τηρ**Αρυ,
where Λ ρ is an A -module.
In this situation arises as usual a spectral sequence (Er, ar), where
/(A, \)®zQP,
E2 = Ext^(A P , Λρ),
determined by the maximal ideal mp C Λ ρ and the induced filtrations.
It turns out that for all ρ > 2 the following holds:
Theorem. The ring Ext-J" (Λ ρ , Λ ρ ) is isomorphic to ExtJ* (Z p , Z p ) , and the algebraic spectral
sequence (Er,dr) is associated with the "geometric" spectral sequence of Adams in the theory H*( , Z p ) .
Here ρ > 2 and A is the usual Steenrod algebra for Ζp-cohomology.
We note that £^** is associated with Ext** (Λ, Λ) ® ZQ (more precisely stated in §12). A
priori the spectral sequence (En ar) is cruder than the Adams spectral sequence in H*{ , Z p )-theory
and £*o** is bigger than the stable homotopy groups of spheres; on account of this, the Adams spectral
sequence for cobordism theory constructed in this work can in principle be non-trivial, since Coo is
associated with Ext / l l / (A j Λ) <g> zQpWe now recall the striking difference between the Steenrod algebra modulo 2 and modulo ρ > 2.
As is shown in H. Cartan's well-known work, the Steenrod algebra for ρ > 2 in addition to the usual
grading possesses a second grading ("the number of occurrences of the Bokstetn homomorphism") of
a type which cannot be defined for ρ = 2 (it is only correct modulo 2 for ρ = 2). Therefore for ρ > 2
the cohomology Ext/i (Z p , Z p ) has a triple grading in distinction to ρ = 2. In §12 we show:
Lemma. There is a canonical algebra isomorphism
Ζ,***
-η
,Τ
Τ
ν
τ-ι
* * *
= Extû (ΛΒ, Λρ) = Ext A (ΖριΖΒ) for / > > 2 .
From this it follows that the algebra E2 for the "algebraic Adams spectral sequence" Er is not
associated, but is canonically isomorphic to the algebra ExtîZp, Z p ) which is the second term of
the usual topological Adams spectral sequence.
If we assume that existence of the grading of Cartan type is not an accidental result of the algebraic computation of the Steenrod algebra A, but has a deeper geometric significance, then it is not out
of the question that the whole Adams spectral sequence is not bigraded, but trigraded, as is the term
From this, obviously, would follow the corollary: for ρ > 2 the algebraic Adams spectral sequence
(Er, ar) coincides with the topological Adams spectral sequence (Er, dr), if the sequence (Er, dr) is
trigraded by means of the Cartan grading, as is {Er, ar). Therefore the orders \n
(SN)\ would coincide with
up to a factor of the form 2h.
Moreover, this corollary would hold for all complexes without torsion (see §12).
830
S. P. NOVIKOV
The case ρ = 2 is more complicated, although even there, there are clear algebraic rules for com-
puting some differentials.
This is indicated precisely in §12.
In this way it is possible not only to prove the nonexistence of elements of Hopf invariant one by
the methods of extraordinary cohomology theory as in [4] (see also § § 9 , 10), but also to calculate
Adams differentials.
The contents of this work are as follows: in §§1-3 we construct the Adams spectral sequence in
different cohomology theories and discuss its general properties.
§§4-5 are devoted to cohomology operations in cobordism theory. Here we adjoin Appendices 1
and 2. This is the most important part of the work.
§§6,
7 are largely devoted to the computations of U* (MSU) and Ext%*u (U* (MSU), Λ).
§ 8 has an auxiliary character; in it we establish the facts from Κ-theory which we need.
§§10, 11 are devoted to computing Ext^*(/(A, Λ).
§§9,
12 were discussed above; they have a "theoretical" character.
Appendices 3 and 4 are connected with the problems of fixed points and the problem of connections between different homology theories from the point of view of homological algebra. Here the
author only sketches the proofs.
The paper has been constructed as a systematic exposition of the fundamental theoretical questions connected with new methods and their first applications. The author tried to set down and in
the simplest cases to clarify the most important theoretical questions, not making long calculations
with the aim of concrete applications; this is explained by the hope mentioned earlier for the role of
a similar circle of ideas in further developments of topology.
§ 1 . The existence of the Adams spectral sequence in categories
Let S be an arbitrary additive category in which Horn (X, Y) are abelian groups for Χ, Υ 6 S,
having the following properties:
1. There is a preferred class of sequences, called "short exact sequences" (0 •-> A^> Β UC -> 0),
such that f'g
= O and also:
a) the sequence (0-> 0^0 ^0-^0) is short exact;
b) for commutative diagrams
A —>B
1 [
A'-*B'
or
Β —>G
\
1
B'->C'
there exists a unique map of short exact sequences
(extending the given square!;
c) for any morphism f: A'-> Β there exist unique short exact sequences 0->C'-»J4'-»B-»0 and
0 H . ^ 4 S - > C - > 0 , where the objects C and C are related by a short exact sequence 0->C"-»0->C-»0
and C and C determine each other.
We introduce an operator Ε in the category S by setting C' = Ε C, or C = EC', and we call Ε the
suspension.
Let Homi(X,Y)
= Hom(X, E^Y) and Horn* (Χ, Υ) = Σ Horn* (Χ, Υ).
831
2. For any short exact sequence 0^,44 B^C -> 0 and any Τ € S there are uniquely defined exact
sequences
T A ) ^ R
i
( T
B ^
, A)
and
-^
which are functorial in Τ and in (0-»/l -*B->C-»0). Here the homomorphisms /*, g.,., /*, g* are the
natural ones and the homomorphisms d, δ are induced by the projection C-*EA in the short exact sequence 0-> Β ->C -> EA->Q according to the above axiom 1.
3. In the category there exists a unique operation of direct sum with amalgamated subobjects:
pairs Χ, Υ G S and morphisms Ζ -> Χ, Ζ -> Υ define the sum X + 2 Υ and the natural maps X-> X +2 Υ
and Υ ^>X +2 y such that the following sequences are exact:
(where Cl and C2 are defined by the exact sequences 0 -> Ζ ->X^> C2 -> 0 and 0 ->Z -> Y-> Cl -»0). By definition we regard X +0Y = X + Υ iwhere 0 is the point object!.
Definition. We call two objects Χ, Υ € S equivalent if there exists a third object Ζ (Ξ S and morphisms f: X^Z
and g: Υ ->Z inducing isomorphisms of the functor Horn* (Z, ) with Horn* (X, ) and
Horn* (Y, ) and of the functor Horn* ( , Z) with Horn* ( , X) and Horn* ( , Y). We call the maps f, g
equivalences.
The transitivity of equivalences follows from the diagram
Χ
Υ
Ζ
ζ +
Η
Τ
γ
τ
where all morphisms are equivalences (by virtue of the axiom on direct sums).
A spectrum in the category S is given by a sequence (Xn, fn), where
fn'. EXn —*- Xn+1
(direct spectrum)
fn'.Xn+ι-^ΈΧη
(inverse spectrum)
By virtue of axioms 1 and 2 in the category S there is a canonical isomorphism
Horn' (X, Y) = Horn' (EX, EY).
Therefore for spectra there are defined the compositions
fn+h-l •••fn'· EhXn-^-Xn+h
fn· • ·fn+h-l'·
Xn+k-*~EhXn
(direct)
(inverse)
which allow us to define passage to the cofinal parts of spectra.
For spectra X = (Xn, /„) a n c j Υ
=
(Υ^
g
J
w e
define
Hom*(X, 7 ) = lim l i m H o m * ( X n ,
Υm)
832
S. P. NOVIKOV
in the case of direct spectra and
Horn* (X, Y)=
lim lim Horn* (X n ,
m
in the case of inverse spectra.
in Horn* ( ,
Ε
Ym)
η
Here, of course, let us keep in mind that in taking limits the grading
) is taken in the natural way.
As usual, remember that the dimension of a morphism
T->Xn is equal to η +η0 — γ, where n0 is a fixed integer, given together with the spectrum, defining
the dimension of the mappings into Xn, and usually considered equal to zero. In addition, Horn and
Ext here and later are understood in the sense of the natural topology generated by spectra.
Thus arise categories S (direct spectra over S) and S (inverse spectra). There are defined inclusions S -> S and S -* S. We have the simple
Lemma 1.1. In the categories
S and S there exist short exact sequences
A, B, C GS or A, B, C C S , satisfying
axiom 1 of the category
0 - * / l - > B - > C ^ 0 , where
S and axiom 2 for the functor Horn* (T, )
if A, B, C C S and Τ 6 S, and axiom 2 for Horn* ( , T) if A, B, C € S and Τ £ S. In the categories
and S there exist direct sums with amalgamation
satisfying
S
axiom 3.
Proof. The existence of direct sums with amalgamation in the categories S and S is proved immediately.
Let us construct short exact sequences in S. Let A, B CS and /: A -> Β be a morphism in S. By
definition, / is a spectrum of morphisms, hence is represented by a sequence Ank~>Bmof
maps.
Consider the set of short exact sequences
(0 ^ Cn ft-> An k-+ Bmk -> 0) and (0 -+ 4 ^ + B^
By axiom 1 of the category S we have spectra in 5, C = (Cn.)
C -> A and Β -> C.
-> C ' ^ - ^ 0) .
and C = ( C ' m ^ ) and morphisms
The corresponding sequences 0 - > C - > / l - > S ^ 0 and 0 -> A -» B -> C -> 0 we call
exact. Since passage to direct limit is exact, we have demonstrated the second statement of the
lemma. For S analogously.
Note that the spectra C and C ate defined only up to equivalences of
the following form: in 5 the equivalence is an isomorphism of functors ΗθΓη*(Γ, C) and Horn* (T, C);
in S, an isomorphism of Horn* (C, T) and Horn* ( C , Γ).
Obviously C = EC.
This completes the proof of the lemma.
Definitions, a) Let X GS.
The functor Horn ( , X) is called a "cohomology theory" and is de-
noted by X*.
b) Let X € S .
The functor Horn* (X,
) is called a "homology theory" and is denoted by X*.
c) The ring Horn* (X, X) for X € S is called "the Steenrod ring" for the cohomology theory X*.
Analogously we obtain the Steenrod ring Horn* (X, X) for X € S (homology theory).
d) The Steenrod ring for the cohomology theory X* is denoted by Ax,
They are graded topological rings with unity.
Αχ.
for the homology theory by
^
Note that an infinite direct sum Ζ = X Z ; of objects Xi CS lies, by definition, in S, if we let
Zn =
Σ Xi
Ax-module,
and Zn -> Zn.i
be the projection. Obviously, Χ* (ΣΑ^) is an infinite-dimensional free
being the limit of the direct spectrum
Horn' (Zn, X) - • Horn* ( Z n + 1 , X),
833
<where X GS C S, all Xt are equivalent to the object X or ΕΎιΧ, and Ε is the suspension.
For an homology theory, if X € S, an infinite direct sum ΣΛ"; is considered as the limit of the
direct spectrum
...-»- 2 -^' -*"
where A ; is ΕΎίΧ,
2
^ i -*-··· J
«~
and therefore lies in S, and the ,4x-module Horn* (Χ, Έ.Χ;) i s free.
By X-free objects for X € S we mean direct sums Σ^f£, where X ; = £ y ' A for arbitrary integers
Ji-
F i n i t e direct sums belong to S.
There are simple properties which give the possibility of constructing the Adams spectral se-
quence by means of axioms 1-3 for the category S.
For any object Τ € S and any X-itee
object Ζ GS we have
Horn* (Γ, Z) = Homl* (Χ* (Ζ), Χ* (Τ)).
Let us give some definitions.
1) For an object Υ € S we understand by a filtration in the category an arbitrary sequence of
morphisms
h
f,
Y=Y.l+-Y0+-Yi^...*-Yi*-....
2) The filtration will be called A^-free for X € S if Zj CS are Λ^-free objects such that there are
short exact sequences
o -»- Yi - t Yi-igX ζ* -> o,
y_i — Y.
3) By the complexes associated with the filtration, for any Τ € S , are meant the complexes
(Cx, dx) and (BT, 8T), where (C x ) £ = X* (Z t) and (θΓ)ί = T*{Zt) and the differentials (9*: ( C J ^ C C * ) ; . ,
and δγ:(Βγ)ί-*(Βγ)
i +1
are the compositions
az:
a n d
δΓ:
X* (Zi) ii- Χ* (Fi-i) - i X* (Zi-t)
5
T.(Zt)-+T.(Yi)K-m~Tt(Zi+i).
4) An A*-free filtration is called acyclic if (Cx, dx) is acyclic in the sense that HO(CX) = X* (Y)
and Ηt(Cx)
= 0 for i > 0.
From the properties (axioms 1 and 2) of the category S and Lemma 1.1 we obtain the obvious
Lemma 1.2. 1) Each filtration (Y "~ Yo *~ Υί *~ . . .) defines a spectral sequence (Er, dr) with term
El = Bj, dl = δγ, associated with ΗοΓη*(Γ, Υ) in the sense that there are defined homomorphisms
0
qa: Hom*(7\ Y) - E ^* , q- Ker qiml -^ E^*, where the filtration (Ker qt) in T*(Y) = Hom*(^ Y) 'is
defined by the images of compositions of filtration maps Γ* (Υί) -> Τ* (Υ).
2) If the filtration is X-free, the complex (Β γ, δγ) is precisely Hom*Ax(Cx,
X* (T)) {with differ-
e n t i a l Horn .χ (dx, l)\.
3) If the filtration is X-free and acyclic, then E1
withExt**AX(X*(Y),X*(T)).
in this spectral sequence coincides
precisely
834
S. P. NOVIKOV
Lemma 1.2 follows in the obvious way from axioms 1, 2 of the category S and Lemma 1.1.
However, the problem of the existence of X-ftee and acyclic filtrations is nontrivial. We shall
give their construction in a special case, sufficient for our subsequent purposes.
Definition 1.1. The spectrum X G S will be called stable if for any Τ GS and any / there exists
an integer re such that Homs (T, Xm) = Homs (T, X) for all m > n, s > /.
Definition 1.2. The cohomology theory X*, X GS, defined by a stable spectrum X will be called
Noetherian if for all Τ G S the Ax-module X* (T) is finitely generated over Ax.
We have
Lemma 1.3. If X* is a Noetherian cohomology and Υ GS, then there exists a filtration
such that Ζ i = Υ iml/Y i is a direct sum Zi =j£j Xn ,· for large rij and the complex C = "ΣΧ* {Z j) is
acyclic through large dimensions.
Here X = (Xn) G S.
Proof. Take a large integer η and consider a map Y—*- £}Xn
such that Χ* ι ^ j Xn)
i
—>-X*(Y)
i
is an epimorphism, where X* is a Noetherian cohomology theory.
By virtue of the stability of the spectrum X, for Υ G S there is an integer η such that the map
Υ -» ΧΧι factors into the composition Υ —^,/j Xn "*~ Zl ElX, where Xn -» X is the natural map. Therex*(foi
*
fore Χ* (ΣΧη .) -> Χ* (Υ)
*-X* (Υ) is an epimorphism. Consider the short exact sequence
i
Obviously X* (Y0(n^)
ô
= KetX* (f0) and Yo
(n>
GS.
Now take a large number n1 »
η and do the same to
as was done to Y, and so on. We obtain a filtration
v
_,
(n)
v
v
(n, η.)
ν
(η.ηι,η,)
where the Z ; are sums of objects of the form ΈιΧπι., with m-i very large.
By definition, C = Έ,Χ* (Ζ;) is an acyclic complex through large dimensions.
Definition 1.3. A stable spectrum X = (Xn) in the category S is called acyclic if for each object
Τ GS we have the equalities:
a) Exti-tAX(X*(Xn),X*(T))=0,
i> 0, t-i<fn(i),
where/•„(»)•*» as π ^ - ;
£
b) Hom^xCY* (JVJ, Ζ* (Γ)) = Hom (r, X) for t < /„, and /„ -» c» a s re ^ ~.
The so-called Adams spectral sequence ( £ r , J r ) with £2-term E2 = Ext*Αχ(Χ*(Υ),
in the following cases:
Χ* (Τ)) arises
1. If in the category S there exists an X-ltee acyclic filtration Υ = Υml *~ Yo ·~ Υ1 *~ •·· *~ Υi-i
*~ Υi •··, on the basis of Lemma 1.2. However, such a filtration does not always exist, since the
theory X* in the category S does not have the exactness property.
2. If Υ G S, Τ GS and the theory X* is stable, Noetherian and acyclic, then, by virtue of
Lemma 1.3, there exists a filtration
835
where the Υ JY ι + l are sums of objects Xn, for numbers η which may be taken as large as we want,
with the filtration acyclic through large gradings. For such a filtration,the corresponding spectral
sequence (Er, dr) has the term E2S'
l
= ExtÂ 5 » '(Χ* (Υ), Χ* (Τ)) through large gradings, by the defini-
tion of acyclicity for the theory X*.
In this way we obtain:
Theorem 1.1. For any stable Noetherian acyclic cohomology theory X £ S and objects Υ, Τ CS,
one can construct an Adams spectral sequence (Er dr), where dr: Εrs·' -» Εrs + r > ' + r ' 1 and the groups
_2J En
are connected to Homm (T, Y) in the following way: there exist
homomorphisms
t—s=m
qf:
Ker qi-i -*• ^ i - j + m ,
i > 0,
where
is the natural homomorphism.
The Adams spectral sequence is functorial in Τ and Υ.
Remark 1.1. The homomorphism g,: Ker q0 -» Ext1'Αχ(Χ*
(Υ), Χ* (Τ)) is called the "Hopf invariant."
Remark 1.2. For objects Τ, Υ £ S and a stable Noetherian acyclic homology theory X € S one
can also construct an Adams spectral sequence (En dr) such that E2 = Ext^^,
this spectral sequence, dT: Erp'
q
-» Erp'r'
q + r +1
(X* (T)y X* (Y)). In
, and the homomorphisms qi are such that
where
q0:Romn(T,Y)^Hom2x (Χ,(Τ),Χ,(Υ))
is the natural homomorphism and Αχ is the Steenrod ring of the homology theory X*.
The proof of Theorem 1.1 is a trivial consequence of Lemmas 1.1-1.3 and standard verifications
of the functoriality of the spectral sequence in the case where the filtration is A^-free and acyclic.
We shall be specially interested in those cases when the Adams spectral sequence converges
exactly to T*(Y) = Horn* (Γ, Υ). Let us formulate a simple criterion for convergence:
(A) If there exists an X-free filtration Υ_i = Υ >- Yo···
*- ΥL (not necessarily acyclic) such that
for any /, I there exists a number i > I, depending on / and I, for which
^j Hom' l (7 7 , F,·) = 0,
then
the Adams spectral sequence converges exactly to Horn* {Τ, Υ). Criterion (A) does not appear to be
the most powerful of those possible, but it will be fully sufficient for the purposes of the present work.
§2. The S-category of finite complexes with distinguished base
points. Simplest operations in this category
The basic categories we shall be dealing with are the following:
1. The S-category of finite complexes and the categories S and S over it.
836
S. P. NOVIKOV
2. For any flat Z-module G (an abelian group such that ® zG is an exact functor) we introduce
the category S ® ZG, in which we keep the old objects of S and let Horn (Χ, Υ) ® zG be the group of
morphisms of X to Υ in the new category 5 ® ZG. Important examples are: a) G = Q, b) G = Q (padic integers). The respective categories will be denoted by So for G = Q and Sp for G = Qp, ρ a
prime.
3. In S (or S p for ρ > 0) we single out the subcategory D (or Dp (_ S p ) consisting of complexes
with torsion-free integral cohomology. It should be noted that the subcategories D and Dp ate not
closed with respect to the operations entering in axiom 1 for S-categories.
These subcategories, however, are closed with respect to the operations referred to, when the
morphism /: A ^B is such that f*: H* (B, Z) -> H* (A, Z) is an epimorphism.
Therefore the category D is closed under the construction of X-itee acyclic resolutions (only
acyclic), and it is possible to study the Adams spectral sequence only for Χ, Υ G D (or Dp).
The following operations are well known in the S-category of spaces of the homotopy type of
finite complexes (with distinguished base points):
1. The connected sum with amalgamated subcomplex X + ?Y, becoming the wedge X\/
Ζ = 0 (a point).
Υ if
2. Changing any map to an inclusion and to a projection (up to homotopy type): axiom 1 of § 1 .
3. Exactness of the functors Horn (X, ) and Horn ( , X).
4. The tensor product Χ ® Υ = Χ χ Υ/Χ V
Υ-
5. The definition, for a pair Χ, Υ €S, of X ®zY> given multiplications X ® Z^X
and Ζ ® Υ -*Y.
6. Existence of a "point"-pair Ρ = (5°, *) such that Χ ® Ρ = X and X ® pY = X ® Y.
-> <— - >
*—
All these operations are carried over in a natural way into the categories So, Sp, S, S, Sp and Sp.
The cohomology theory X* will be said to be multiplicative if there is given a multiplication
*-v XT\r
A. —^~ Λ.,
Z Qy
γ
Λ. t
rr
Ο,
The cohomology theory Y* is said to act on the right [left] of the theory X* if there is given a
multiplication X
®Y^XotY®X^X.
The previously mentioned theory P*, generated by the point spectrum Ρ = (S°, *), operates on all
cohomology theories and is called "cohomotopy theory." Its spectrum, of course, consists of the
spheres (S"). It is obviously multiplicative, because Ρ ® Ρ = P.
We now describe an interesting operation constructed on a multiplicative cohomology theory X = (Xn)
€ S of a (not necessarily stable) specCrum of spaces.
Let (//„ [ ) be the spectrum of spaces of maps //'„ = Qn'iXn = Map(S"-', Xn).
tive and Ρ ® Ρ = P, we have a multiplication
Since X is multiplica-
Η η Χ Hm-*~Hm+nLet now ί = / = 0. Then
η η A H τη ~*" •" m+n ·
Suppose that the cohomology ring X* (X) and all X* (K) have identities (the cohomology theory con-
837
tains scalars with respect to multiplication X ® X-^X). Consider in the space H°n the subspace
Hn C H°n ^Ω^λ",, which is the connected component of the element 1 GX°(P). We have a multiplication
Η η Χ Ηη -> Η η
\
1
Ι
Η°ηΧΗ°η^Η°η
induced by the inclusion Hn CH°n.
Let π {Κ, L) be the homotopy classes (ordinary, non-stable) of maps Κ -> L, and let Π" 1 (Χ) =
lim n(K, //„). Obviously Π" 1 (K) is a semigroup with respect to the previously introduced multiplican->oo
tion. We have
Lemma 2.1. Π" 1 (Κ) is a group, isomorphic to the multiplicative group of elements of the form
\l + x\ € X° {K), where χ ranges over the elements of the group X° (K) of filtration > 0.
The proof of Lemma 2.1 easily follows from the definition of the multiplication Hn χ Ηm->//m
by means of the multiplication in the spectrum X.
+ n
Therefore the spectrum (Hn) defines an "//-space" and the spectrum BH = {BHn) has often been
defined. The set of homotopy classes π{Κ, BH) = \imn{K, BHn) we denote by Π° (Κ), while Ώ°(ΕΚ)
= Π " 1 (Κ) by definition, where Ε is the suspension.
The following fact is evident:
If Κ = E2L, then Π 0 (K) = X1 (K); therefore in the S-category Π 0 (K) is simply X1 (K). As we have
already seen by Lemma 2.1., this is not so for complexes which are only single suspensions, where
n°(£L) consists of all elements of the form {l + x\ in X° (L) under the multiplication in X" (L).
An important example. Let X = Ρ = (Sn, *). Then the spectrum Η η with multiplication //„ χ //„
-» Η n is homotopic to the spectrum Hn (maps of degree +1 of Sn -» Sn with composition Η η χ Hn -» //„).
The /-functor of Atiyah is the image of K(L) -> Π 0 (L) in our case X = P. In particular, in an S-category L = E2L' we have that ΓΡ (L) is P* (L); in the case L = EL', Π 0 (L) depends on
the multiplication in P* (L').
Besides the enumerated facts relating to the S-category of finite complexes one should also mention the existence of an anti-automorphism σ: S^>S of this S-category which associates to a complex
X its S-dual complex (complement in a sphere of high dimension). The operator σ induces
σ: 3 - > 5 ,
σ: ~S-+S,
2
σ = 1.
Since Hom(Z, Y) = Horn (oY, σ X) and σΧ is a homology theory in S if X is a cohomology theory, then
the duality law of Alexander — Pontrjagin is, obviously, the equality Χ* (Κ) = σΧ*(σΚ), and by an
λ^-homology manifold is meant a complex Κ such that X1 (K) = aKn_i(K) in the presence of some natural
identification of σ Χ* (Κ) with σΧ*. (σ Κ); for example, if Κ is a smooth manifold, then σ(Κ) according
to Atiyah L"J is the spectrum of the Thorn complex of the normal bundle in a sphere. In the presence
of a functorial Thom isomorphism in X*-theoty for some class of manifolds, we obtain Poincare-Atiyah
duality.
Let X G S,, Υ 6 S, Γ € S. In V 1 we constructed the Adams spectral sequence with E2 term equal
to Ext7*
(X*(Y),X*(T)).
838
S. P. NOVIKOV
The law of duality for Adams spectral sequences reads:
The cohomology Adams spectral sequence (Er, dr) with term E2 = Ext^^ (Χ* (Υ), Χ* (Τ)) is canonically isomorphic to the homology Adams spectral sequence (E'r, d'r) with term E'z = Ext / i c r A (σΧ*(σΥ), σΧ* (σ Τ)). The homology Adams spectral sequence for X = σ Χ = Ρ was investigated by
A. S. Miscenko [16].
Let us introduce the important notion of (m-l)-connected spectra.
Definition 2.1. The spectrum (Xn, fn) = X (direct) is called (m — l)-connected if each object Xn is
(ra +m — 1 +raQ)-connected, where the integer n0 is defined in § 1 . Analogously for inverse spectra.
Usually n0 = 0 and Xn is (n + m — l)-connected, /„: EXn -> Xn
for inverse.
+l
for direct spectra.
Analogously
Finally, we should formulate two obvious facts here, which will be used later.
->
Lemma 2.2. a) If X G S, the cohomology theories EX and X have the same Adams spectral se->
<quences for any Υ and Τ for which the sequences exist (here Υ € S, Γ € S).
b) Furthermore, if X = Σ Ε lXis a direct sum, where γ. -» oo for i -> oo, fAere iAe theory X* defines
the same Adams spectral sequence as the theory X*.
Proof. Since each X-ttee acyclic resolution is at the same time an Z-free resolution, the lemma
at once follows from the definitions.
From the lemma follows
Corollary 2.1. For any stable Noetherian acyclic cohomology theory X € S and any Υ £ S and
Τ eS, all groups Εχί^χ(Χ* (Y), X* (T)) ® ZQ = 0 for s > 0.
Proof. Since a stable spectrum X in the category So = S ® zQ l s equivalent to a sum Σ ΕΎιΚ(Ζ)
of Eilenberg-MacLane spectra for π - Ζ, and since for X' = K(Z) the ring A ® %Q l s trivial, it follows that all Ext s χ( , ) (g) zQ = 0 for s > 0, since ExtÂ"' £5 zQ ( > ) = 0 for s > 0 and by virtue of
Lemma 2.2.
§3· Important examples of cohomology and homology theories.
Convergence and some properties of Adams spectral sequences
in cobordism tiieory
We list here the majority of the most interesting cohomology theories.
1. X = Κ (π), where Xn = Κ (π, η). This theory is multiplicative if 77 is a ring, and X = Η ( , π).
The case π = Zp is well known, having been studied in many works L ' ' ' ' J. The spectral
sequence was constructed by Adams in L J, where its convergence was proved (π = Ζρ). The ring A
is the usual Steenrod algebra over Z p . Here the commonly studied case is ρ = 2. The case ρ > 2 was
first studied in [ J .*
The criterion (A) for the convergence of the Adams spectral sequence applies easily in the category Sp = S <S> zQp under the condition that Υ is a complex with ττi * (Υ) ® ZQP finite groups, in
which case there is a nonacyclic resolution (the Postnikov system) which is X-ftee.
*In Theorem 2 of the author's work L^^J there are erroneous computations, not influencing the basic results. We note also the peculiar analogues, first discovered and applied in L 2 ^J, to the Steenrod powers in the
cohomology of a Hopf algebra with commutative diagonal. It turns out that for all ρ > 2 these "Steenrod powers"
Stp1 are defined and nontrivial for i = 0, 1 mod (p-1), i > 0. These peculiar operations have never been noted in
more recent literature on these questions, although they are of value; for example, they reflect on the multiplicative formulas of Theorem 2 in L^J for ρ > 2.
839
In the case π = Z, as is easy to see, the applicability of criterion (A) in the category S itself
again easily follows from the properties of the usual contractible spaces and Postnikov systems (see,
for example, [16J).
2. Homotopy and cohomotopy theories. Let Ρ be the point in S, where Ρn = S". The theory P* is
that of stable homotopy groups, and P* that of stable cohomotopy groups. The (Eckmann-Hilton) dual
of this spectrum is Κ (Ζ) and the theory H* ( , Z). Similarly, the spectra P ( m ) = P/mP (m an integer)
are Eckmann-Hilton duals of the spectra K(Zm).
For the homology theory P* (X) the proof of convergence of the homology Adams spectral sequence
with term E2 = Ext*^ is similar to the proof for the cohomology spectrum K(Z) by virtue of EckmannHilton duality and follows from criterion (A) of § 1 .
The proof of convergence for the theory Ρ(m)* analogously proceeds from the method of Adams
for K(Zm).
These theories were investigated in L ^ J .
By virtue of the law of duality for the Adams spectral sequence (cf. §2) and the fact that σ Ρ = Ρ
and η P(m) = Ρ(m), we obtain convergence also in cohomotopy theory, where σ is the S-duality operator.
3. Stable K-theory.
a) Let k = (kn), where il2nk2rl = BU χ Ζ,and the complexes kn are (re — l)-connected. Then k2n is
the (2re —l)-connected space over BU and the inclusion x: k2rt-*k2n.2
is defined by virtue of Bott periodicity.
Here kl = Kl for i < 0 for K* the usual complex K-theory, and if H* (L, Z) has no torsion, then
k2l(L) is the subgroup of K2l(L) consisting of elements of filtration > i.
b) Let kO = (kOn), where Wnk0sn
= (.k0j-lh,
= BO χ Ζ, and all k0n are (n-l)-connected. We have kO ^
where nSnkoJ-n1.} = BO χ Ζ, kO^
= W and the kO J ^ are (n-l)-connected. Here i is
to be taken mod 8.
It is easy to show that in the category S <8> χΖ [l/2] all spectra kO
and the spectrum It is a sum of two spectra of the type k = kO + E2kO
.
coincide up to suspension,
4. Cobordism. Let G = (Gn) be a sequence of subgroups of the groups 0 a ( ( l ) , where a(rc + 1)
> a(re) and a{n) -» °o for η -» ~, with Gn C Gn + 1 under the inclusion 0a
C 0 a ( n + 1 ) . There arise
natural homomorphisms BGn -> BGn +l and a direct spectrum (not in the S-category) BG. With this
spectrum BG is connected the spectrum of Thom complexes MG = (MGn) in the category S.
Examples:
a) The spectrum G = (e), e C 0n; then MG = Ρ;
b) G = 0, SO, Spin, U, SU, Sp; then MG = MO, MSO, «Spin, MU, MSU, MSp have all been investigated. All of them are multiplicative spectra and the corresponding cohomology rings have commutative multiplication with identity. Let us mention the known facts:
2) MS0®z<?2 = S ΕλίΚ(Ζ) + Σ Εμ«Κ(Ζ2) [CM. («),
5
3)
MG ®zQP
=ΣΕ>
k
840
S. P. NOVIKOV
where Η (Λϊ(ρ), Z p ) = Α./βΑ +Αβ, A is the Steenrod algebra over Z p and β is the mod ρ Bokstefr
homomorphism. This result holds for G = SO, U, Spin, Sp forp > 2, G = U for ρ > 2, and G = Si/ for ρ > 2
with reduction of the number of terms λ 4 corresponding to certain partitions ω (see [ *5, 17, 18, 26]).
4)
Μ Spin ®z<?2 = Σ ^
(Ζ*) + Σ # μ *Α0 + Σ
Facts (1) and (2) are known, and fact (4) is given in a recent result of Anderson-Brown-Peterson
[10].
c) G = T, where Tn = Gn C Un C02n is the maximal torus. This leads to MG, again a multiplicative spectrum since MTm + n = MTm (g> MTn.
Let us mention the structure of the cohomology M*(p )(/>), w here Ρ = (S0,*) is a point, M*(p)(P) =
Qp [xi, . . . , Xi, . • .] (polynomials over Qp) with dim*; = —2pi +2 and fW°p)(P) the scalars (? p .
The ring U* (P) for G = {/ (spectrum Λίί/) is Ζ [y,, . . . , y £, . . . ] , where dim y £ = — 2i.
For the spectra 'W(p) = Ζ and MU = X we have the important, simply derived
Lemma 3.1. If a £ A is some operation for X = M( p ) 6 S ® zQp or X = MU G S which operates
trivially on the module X* (P), then the operation a is itself trivial.
Proof. Since ο € Horn* (X, X), the operation a is represented by a map X' -* Ε ' X. Since π* (Χ)
® zQp a n d H* (X, Qp) for X = Λ/(ρ) and X = M(7 do not have torsion, it follows from obstruction theory
in the usual fashion that the map a:XÊyX
is completely determined by the map α*: π* (Χ) -> π* (Χ),
which represents the operation a on X* (P), for X'l(P) = π^(Χ). End of proof of lemma.
Since MU ®zQp ™ ^J Ε hM(ρ)
, we have the following fact:
where X = MU, X* = U*, and Κ = M(ph Y* = i/* ( p ) = M* ( p ) ; we denote Af*(p) by t / * ( p ) and Mi/* by i/*.
Both are multiplicative theories. This fact, that the Ext terms and more generally the Adams spectral
sequences coincide, follows from the fact that MU ®zQp = - 2 ExkM(V), as indicated in §2, since
MU <S> zQp is
a
s u m
k
°f suspensions of a single theory Λ/(ρ) and Qp is a flat Z-module.
For any multiplicative cohomology theory X* there is in the ring A the operation of multiplication by the cohomology of the spectrum P, since the spectrum Ρ acts on every spectrum: Ρ ® X = X.
In this way there is defined a homomorphism X* (P) -> 4 , where X* (P) acts by multiplication. From
now on we denote the image of X* (P) -» A by Λ C A , the ring of "quasiscalars."
For spectra Ζ = Λ/(ρ), ^ = Λ/£/ we have the obvious
Lemma 3·2. Lei Κ € D p be a stable spectrum. Then X* (Y) is a free Α-module, where the minimal
dimension of the Α-free generators is equal to n, if Υ = (Υm) is a spectrum of (n + m)-connected complexes Ym.
The lemma obviously follows from the fact that in the usual spectral sequence in which E2=H*(Y,
Χ* (Ρ)) = Η*(Υ, Λ) for X = M(ph MU, all differentials dr = 0 for r > 2, and the sequence converges to
X*(Y).
841
Now let Υ satisfy the hypotheses of Lemma 3-2. We have
Lemma 3.3. There exists an X-free acyclic resolution for X = M{p), MU: Υ <- Yo *- y,*~ . . . *~
Υ l ·- . . . , where the stable spectra Υ £ 6 D p ore (m + 2i — l)-connected, if } is a stable (m — 1)connected spectrum in Dp. Furthermore, if X = M{p), the spectrum Υ ι is (m + 2i (p — 1)— \)-connected.
Proof. Since F is an (m— l)-connected stable spectrum, the minimal Λ-free generator of the module
X* (Y) has dimension m, and the set of m-dimensional Λ-free generators corresponds to the generators
of the group Hm(Y, Qp).
Choose in correspondence with this system of Λ-free generators an X-free object Co and construct
in a natural way a map /„: Υ -> Co such that
fo.:Hm+h(Y,
Qv)-+Hm+h (Co, Qp)
is an isomorphism for k < 1. Obviously Co is also (m —l)-connected. Then the object YQ such that
0 -> F o -> Υ -» Co -» 0 is a short exact sequence has the property that it is also a stable spectrum in Dp.
Furthermore, since f0* is an isomorphism on the groups Hm +/C(Y, Qp) for k < 1, the object Yo is mconnected in Dp. If X = iW(p), then it may be shown furthermore that in constructing Co in correspondence with Λ-free generators in X* (Y) the map / 0 *: Ηj{)f, Qp) -> Η;(C0, (? p ) is an isomorphism for
/ < m +2p—3 and a monomorphism for j = m + 2p —2.
Therefore F o will be (m + 2p — 3)-connected if Υ is (m — l)-connected. The result for X = MU in the
category D is obtained by substituting the minimal ρ = 2. This process we continue further, and obviously obtain the desired filtration. The lemma is proved.
Now let Τ €. S be a finite complex. By virtue of Lemma 3.3 we have that Ηοπι'(Γ, Υ j) = 0 for
large i. Therefore the Adams spectral sequence converges to Ηοπι*(Γ, Υ) by virtue of criterion (A)
in§l.
From these lemmas follows
Theorem 3·1· For any stable (m — l)-connected spectrum Υ € D C S, X = MU and any finite complex
Τ € S of dimension n, the Adams spectral sequence (Er, dr) with term E2 = Exttx(/V* (Υ), Χ* (Τ))
exists and converges exactly to Horn* (T, Y); moreover E x t . £ (X* (Y), X* (T)) = 0 for t — s < s + m —re.
Furthermore, the p-primary part Ext^x' (X* (Y), X* (T)) ® zQp = 0 for t < 2s (p-1) + m-n.
The proof follows immediately from the fact that if Τ is an re-dimensional complex and Υ is a kconnected spectrum, then Hom^x (Χ* (Υ), Χ* (Γ)) = 0 for i < k — η and from Lemma 3.3 for X = MU.
The statement about the p-components of the groups Ext follows from Lemma 3.3 for the spectrum
, since
M(p)
(p),
The theorem is proved.
Note that for X = MU, M(p), stable spectra Υ and finite complexes T, all groups E x t 5 ' ^
sion groups for s > 0, as derived in §2.
are tor-
Let X = M (p ), y € Dp be a stable spectrum, and Τ € S ® zQp, where the cohomology Η* (Υ, Qp)
and H* (T, Qp) is different from zero only in dimensions of the form 2k(p — 1).
842
S. P. NOVIKOV
Under these hypotheses we have
Theorem 3-2. a) The groups Hom^x (Χ* (Υ), Χ* (Τ)) are different from zero only for i = 0 mod 2p —2;
b) Ax
is a graded ring in which elements are non-zero only in dimensions of the form 2k(p—l);
c ) The groups Ext S'J(X* (Y), X* (T)) are different from zero only for t = 0 mod2p-2;
A.
d) In the Adams spectral sequence (Er, dr) all differentials
dT are equal to zero for r φ 1 mod 2p —2.
Proof. Since the ring X*(P) (P a point) is nontrivial only in dimensions of the form 2k(p — 1), statement (b) follows from Lemma 3.1. Statement (a) follows from (b) and the hypotheses on X* (T). From(b)
it follows that it is possible to construct an Ax-free acyclic resolution for X* (Y) in which generators
are all of dimensions divisible by 2p—2. From this (c) follows. Statement (d) comes from (c) and the
fact that dr(Ef'')CEsr
+ r
·
i + r
- ' . Q.E.D.
Corollary 3.1. ForX = MU, Υ = Ρ, Τ = P the groups E x t ^ ' (Χ* (Ρ), Χ* (Ρ)) ®zQp
t < 2s(p—1) and for f ^ 0 mod2p — 2, and the differentials
dr on the groups Er ® zQp
= Q for
a r e
equal to zero
for r ^ 1 mod 2p —2.
From now on we always denote the cohomology X* for X = MU by U* and the Steenrod ring Ax
by
In the next section this ring will be completely calculated.
AU.
As for the question about the existence of the Adams spectral sequence in the theory U* and
category S, we have
Lemma 3.4.
The cohomology theory U* is stable, Noetherian,
and acyclic.
Proof. The stability of the spectrum MU = (MUn) is obvious.
Let Γ be a finite complex. We shall
prove that U* (T) is finitely generated as a Λ-module, so of course as an A -module, where A = U* (P)
C A . Consider the spectral sequence (En dr) with term E2 = Η* (Τ, Λ), converging to U* (T). Since
Γ is a finite complex, in this spectral sequence only a finite number of differentials dr, . . . , dk are
different from zero, d^ = 0 for i > k. Note that all dr commute with Λ, and Εχ
ciated with U* (T), where Εχ
= Εk.
as a Λ-module is asso-
The generators of the Λ-module E2 lie in Η* (Τ, Ζ); A" = Ζ and
they are finite in number: ϋ,(Γ), . . . , ujr)
€ E*·".
Note that dr(Epr· q) C £? +r- «*r + 1 . Denote by Λ^
C Λ the subring of polynomials in generators of dimension < 2/V, Λ = U* (P) = Qy.
Noetherian. Similarly, let Λ
viously, Λ = Λ/γ 0 2.K
The ring Λγ is
C Λ be the subring of polynomials in generators of dimension > 2/V. Ob-
and Λ has no torsion.
Assume, by induction, that the Λ-module Er has a finite number of Λ-generators u\j , . . . , ^ / r '
r
and there exists a number ΝΓ such that Er = Er ® ^Λ
number of generators u ^ , . . . , u\r),
dim λ, .
(r)
< /Vr for all k, j.
Set Nr+1
erian property of the ring ANr
t
, where Εr is a Λ^ -module with the finite
above. Consider dr(,Ujir) ) = Σ kûkir),
= Max (/Vr, Nr).
Then λ^,
(Γ)
r +l
Nr
, the module Η (Er ® zA
6 ΛΛ
where λ^° G Λ. Let
By virtue of the Noeth-
is generated by polynomial generators of dimension Nr < k < Nr + i and ANr
Since
Γ
ι
+ <8> χΛ
r
1
+ =Λ
N
H(Er, dr) = Er+i = Η (Er ®zAN *, dr) = H{Er ®zANrr+i
+1
I
if we set ΕΓ + ι = Η (ΕΓ ® Λ ^ Γ
Γ
r+1
is finitely generated, where ANr
,dr)
® AN r+i, dr)
N
+ 1 , dr), then £ r + 1 is a finitely generated A/vr + i -module, and Er
+l
Γ
.
METHODS OF ALGEBRAIC TOPOLOGY FROM CORDISM THEORY
84
3
Taking N2 = 0, we complete the induction, since for some k, Εk = Εχ is a finitely generated
Λ-module. Therefore the module U* (T) is finitely generated and the theory U* is Noetherian.
Let us prove the acyclicity of the theory U* in the sense of § 1 . Since the (4n—2)-skeletons X2n
of the complexes MUn do not have torsion, by virtue of the lemma for these complexes in the category
D the spectral sequence exists; moreover, the module U* (X2n) is a cyclic A -module with generator
of dimension 2re and with the single relation that all elements of filtration > 2re in the ring A annihilate the generator. From this and the lemma it follows that Ext l'J (U* (A 2n )) = 0 for t < In— dim T,
and Hom*Au (U* (X2n), ) = Horn* ( , X2n) = Horn* ( , X) in the same dimensions. From this the lemma
follows easily.
Lemma 3.4 implies
Theorem 3.3. For any Υ, Τ € S there exists an Adams spectral sequence (Er, dr) with term Er
=
Ext*A*u(U*(Y),U*(T)).
A. S. Miscenko proved the convergence of this spectral sequence to Horn* (T, Y) (see L^"J).
§4. O-cobordism and the ordinary Steenrod algebra modulo 2
As an illustration of our method of describing the Steenrod ring A (see § § 5 , 6) we exhibit it
first in the simple case of the theory 0*, defined by the spectrum MO isomorphic to the direct sum
MO = 2 ΕλωΚ(Ζ2),
where ω = (o 1 ; . . . , o s ) Σα; = λω, at φ 2 ; - l , or ω = 0. The Steenrod ring A0
ω
is an algebra over the field Z 2 . Let A be the ordinary Steenrod algebra. The simplest description of
the algebra A is the following: A = GL (A) consists of infinite matrices a = (αω ω), where ω, ω'
Site nondyadic partitions (a,, . . . , a s ) , (a[, . . . , α^), α ω , ω' €• A and dim a = λω — λ ω ' + dim αω< ω' is
the dimension of the matrix.. The ring GL (A) is, by definition, a graded ring. This describes the
ring A more generally for all spectra of the form ΈΕ ωΚ(Ζ2).
In the ring GL {A) we have a projection operator π such that π Α π = Α, π2 = 1, π € A0 = GL{A).
Another description of the ring A is based on the existence of a multiplicative structure in
0* (K, L). Let Λ = 0* (Ρ) = Ω0 be the unoriented cobordism ring, 0\P) = illQ.
1. There is defined a multiplication operator
X-+OX, xeE0*(K,L),
o e A = O'(P).
This defines a monomorphism Λ -> A .
2. We define "Stiefel-Whitney characteristic classes" Ψ^ξ) € 0l (X), where ξ is an 0-bundle
with base X:
a) for the canonical 0,-bundle ζ over RP we set:
Wi(l)=0, ΐΦΟ,Ι,
η large, D the Atiyah duality operator.
b) If η = £ Θ ξ2, then W (η) = W (ξ,) W (ξ2), where W = Σ Wt.
These axioms uniquely define c l a s s e s If; for a l l 0-bundles.
As usual, the c l a s s e s Wt define c l a s s e s Wω for a l l ω = (α,, . . . , a s ) such that Wι = W^
,).
In O-theory there i s defined the Thom isomorphism φ : 0* (X) — 0* (Μζ, * ) , where Μξ is the Thorn com-
844
S. P. NOVIKOV
plex of ξ. L e t X = B0n, Μξ = M0n. L e t u = φ (I) € 0*(M0n).
We define operations
f, L) ->- Oi+d^{K, L)
by setting S q ' » = <£ (IFJ, where ίτω € 0* (B0 n ).
Under the homomorphism ϊ •]* : 0* (MOn)-+ 0" (BOn)-+ 0* fj]' ΛΡΑ°° j
the element u =
Λ*
goes into i* /* (it) = ii, . . . « „ , where u£ € 0 1 (RP°°) is the class Wl (£;), £. the canonical 0,-bundle
over RP°° , defined above, and Sq^ (Ui . . . un) = Sw (u^ .. . , un) ul . . . un, where SM is the symmetrized
monomial Eu" 1 . . . u"s, s < n.
There is defined the subset Map (A", MO,) C O 1 ^ ) and a (non-additive) map y: O1 (X) -* Hl(X, Z2)
-» Map(/i, MO,), where e : 0* ->ff*( , Z2) is the natural homomorphism jdefined by the Thom class!.
The operations Sq w have the following properties:
a)
Sq°(xy)=
Σ
Sq«>(x)Sq»>(y);
(ωι, ύ>2)=ω
b) if χ = γ(χι),
then Sq^ (x) = Ο, ω ^ (λ) and Sq A (*) = χ έ +1 ;
c) the composition Sq ' oSq 2 is a linear combination of the form SA^jSq", λ ω C Z 2 ) which can
be calculated on u = 0(1) € 0* (M0n) or on t* /* (u) =.u, . . . un G 0* (RP™ χ . . . χ /?Pn°°), u ; € Im γ;
d) there is an additive basis of the ring A of the form SAâ^Sq™', λ; €Ζ2, ai an additive basis
of the ring Λ = 0* (Ρ) = O Q . Thus A is a topological ring with topological basis a^Sq", or
A° = (A-S)A,
where / \ means completion and S is the ring spanned by all Sq&).
We note that the set of all Sq" such that ω = (α,, . . . , as), where a, = 2' — 1, is closed under composition and forms a subalgebra isomorphic to the Steenrod algebra A C S C A .
How does one compute a composition of the form S q " ο α, where α G Λ ? We shall indicate here
without proof a formula for this (which will be basic in § 5 , where the ring A is computed).
Let (Χ, ζ) be a pair (a closed manifold and a vector bundle <f), considered up to cobordism of
pairs, i.e. (Χ, ξ) £ 0* (BO). In particular, if ζ = ~rx, where τχ is the tangent bundle, then the pair
a, i ) e o o =o*(P).
We define operators ("differentiations")
by setting Ψ*ω (Χ, ξ) = (Υω, [*ω(ξ+Τχ)-τγ
), where (Υω, / „ : F ^ X ) is D ^ (£) G 0*
We also have multiplication operators
a: 0,(B0)^0,(B0),
α: Ω 0 ->Ω 0 ,
845
where (Χ, ξ) - (Χ χ Μ, ξ κ (~τΜ)) and (Μ,-τΜ) € Ωο represents α €Ω0.
In particular, we have the formula
s w:t(a)-w:2,
w:-a=
ω=(ωι, ω?)
where α ε Ω 0 , ^ ( α ) 6 ί 1 ο .
It turns out that the following formula holds:
Sq*.a=
where a € Λ = U,Q.
We also have a diagonal
where Δ (α) = α (8) 1 = 1 <8> α, and ASq01 =
2
δ( ω
ϊ ' ®S^2'
s o t h a t
Λ ° <8> Ωο Λ 0 may be con-
ω=(ωι, ω2)
sidered as an /l°-module via Δ; /1° <8> Ωο ,4° = 0* (MO ® Λ/0), and Δ arises from the multiplication
in the spectrum, MO ® MO ^ MO.
We note that the homomorphisms Ψ*ω coincide with the Stiefel characteristic residues if η = dimco.
We also note that any characteristic class h € 0* (SO) defines an operation h €. A , if we set
h(u) = <£(A), where u € 0* (MO) is the Thorn class and φ: 0* (BO) -* 0* (MO) is the Thorn isomorphism.
In particular we consider the operations
d(u) = ψ (hi), where h± =
Δ ; (α) = φ(/* 2 ), where h2 =
y(Wi),
y(Wi)2.
It turns out that d2 = 0, Ad = 0 and the condition Α, (ξ) = 0 defines an SO-bundle, since h1 = γ($\).
Further, it turns out that 0* (MSO) is a cyclic A -module with a single generator υ d 0* (MSO),
given by the relations d(v) = 0, Δ (υ) = 0, and we have a resolution
(
where Co = A
^CWcAo*(MSO)-+0)=C,
^Ci^-
(generator uo),C£ = A
+ A (generators ui, vit i > 1), and
d(uj) = dui-i, i ^ 1,
d(vt)= Aui-i.
The homomorphisms d* and Δ*: Ωο -» Ωο coincide with the homomorphisms of Rohlin [ 2 0 ] , [ 2 1 ] and
Wall [ 2 3 ] .
We consider the complex Horn* AQ (C, 0* (P)) with differential d* defined by the operators d* and
Δ* on 0* (P) = Ω . The homology of this complex is naturally isomorphic to Ext** 0 (0* (MSO), 0* (P))
or the E2 term of the Adams spectral sequence.
It is possible to prove the following:
846
S. P. NOVIKOV
1) all Adams differentials are zero;
2) Ext0 ·*Αθ(Ο*(Μ SO), 0* (P)) = Ω So/2nso
tion of the complex C;
3) E x t ' ' ^
1
+ S
4) Ext ' ^
+4/t
C Ωο where Ω5Ο/2Ω5Ο = Kerd* ft KerA* by defini-
= 0, for s jt 4k;
(0* (MSO), 0* (P)) is isomorphic to Z2 + . . . +Z2, where the number of summands is
A.
equal to the number of partitions of k into positive summands (A,, . . . , ks), ΣΑ£ = k;
5) there exists an element h0 € Ext 1 '^ associated with multiplication by 2 in Εχ, such that
0 1
Ext ' ., ->° Ext 1 ' t r j M is an epimorphism, t = 4k, and Ext'· ' ^° Ext 1 + 1 ' ! t ' i s an isomorphism, i > 1.
Α
Α
Α
Ά
**"
These facts actually are trivial since
Ext A o (0* (X), 0* (7) = Ext A ( # ' (X, Z2), J?* (Y,
and H*(MSO, Z2), as was shown by the author t 1 ? , 18]
a n d
b y
Wall [23],
Zi)),
i s
H* (1E'K(Z2))
+ H*
(ΣΕ K(Z)), where there are as many summands of the form K(Z) as would be necessary for (4) and (5).
We have mentioned these facts here in connection with the analogy later of MSO with MSU and the
paper of Conner and Floyd [13].
In the study of Ext AU (U* (MSU)) all dimensions will be doubled, the groups E^sk+l
for 1 < i < 3
will be constructed in an identical fashion, but the element h0 € Ext 1 ' 1 will be replaced by an element
h € Ext1'^, and the Adams differential d3 will be non-trivial (see § § 6 , 7).
We note that the construction described here gives us a natural representation of the ring A
= (A°S) on the ring Ωο by means of the operators Ψ*ω ("differentiation") and the multiplication on Λ .
In a certain sense the operators Wω generalize the ordinary characteristic numbers. They can be
calculated easily for [RP2h] € Ωο and
(ωι, ω 2 ) = ω
(the Leibnitz formula). Completing their calculation would require that they be known also for "Dold
manifolds."
It is interesting that the ring A C A , where A C S, is also represented monomorphically by the
representation Wω on Ω^.
In conclusion, we note that the lack of rigor in this section is explained by the fact that 0*theory will not be considered later and all assertions will be established in the more difficult situation
of U*-theory.
§5. Cohomology operations in the theory of t/-cobordism
In this section we shall give the complete calculation of the ring A of cohomology operations in
U*-cohomology theory. We recall that for any smooth quasicomplex manifold (possibly with boundary)
there is the Poincare-Atiyah duality law
U*(X) =<Un-i{X,dX)
and Ui(X)
=
ϋη-*(
where quasicomplex means a complex structure in the stable tangent (or normal) bundle. Here there
is also the Thom isomorphism φ: Ul (X) -> U2n + ' (Μξ, *) where ζ is a complex f rt -bundle of dimension
847
2n, and Μξ is its Thom complex. We denote the Poincare-Atiyah duality operator by D. There is
defined a natural homomorphism e : £/* (X) -> i/* (P), where Ρ is a point and Q,y = f/*(P) = Ζ [Ϊ,, . . . ,
We consider the group i/* (X) given by pairs (A!, /), where I is a manifold and /·' X->K. Let α be
an arbitrary characteristic class, α € ί/* (BU). For any complex Κ in the category S, the class α defines an operator
a: U
if we set a(X, {) = (Ya, f-fa),
where (Y a, fa) C i/* (A!) is the element having the form D α(-τχ),
where τ χ € Κ (Ζ) is the stable tangent t/-bundle of X. \Da(-rx)
= D((~rx)*
(a)).\
As we know, the operation of the class α on ί'* (Κ) can be defined in another way: since V* (MU)
= U* (BU) by virtue of the Thom isomorphism φ, we have φ (α) = α (Ξ U*(MU) = A . We consider the
pair L = (K \J P, P) in the S-category; then i/* (K) = Horn* (P, MU ® L) by definition, where Ρ is the
spectrum of a point. Every operation a = φ (a) defines a morphism φ(α):Μυ ^>MU and, of course, a
morphism
φ(α) <g> 1:
MU®L-+MU®L.
Hence there is defined a homomorphism a*:U*(K) -*U*(K) by means of φ (a) ® 1. We have the simple
Lemma 5.1. The operators a* and a* coincide on U*(K).
The proof of this lemma follows easily from the usual considerations with Thom complexes, connected with t-regularity.
Thus there arises a natural representation of the ring A
on ί/* (Κ) for any K, where a -> ι φ'1 (a)]*
= α*, φ-.υ*(Βυ)^υ*(Μυ).
We have
Lemma 5.2. For Κ = Ρ, the representation a -> [φ" (α)]* of the Steenrod ring AU in the ring of
endomorphisms of U*(P) = Ω;/ is dual by Poincare-Atiyah duality to the operation of the ring A on
U* (P) and is a faithful
representation.
Proof. Since Κ = Ρ and MU <8> Ρ = MU, the operation of the ring Au on Horn* (P, MU) is dual to
the ordinary operation, by definition. By virtue of Lemma 3·1 of § 3 , this operation is a faithful representation of the ring A . The lemma is proved.
We now consider the operation of the ring A
on t/* (P) and extend it to another operation on
t/* (BU). Let χ € U*(BU) be represented by the pair (Χ, ξ), ξ&Κ°(Χ).
1
We set
where a = φ' (a), a eU* (MU) and (Ya, fa) is the element of U* (X) equal to Όα(ξ),
α € 6'* (Βί/), and
Γ
Μ is the stable tangent (/-bundle of M.
if ζ = TX, then f*a (ξ + τχ)-τγa = -τΥα
and hence the pair (Χ, -τχ) goes to (Υα, -τγα), i.e., the
subgroup U*(P) C ί/* (βί/) is invariant under the transformation ο .
We have the obvious
Lemma 5.3· ΤΆβ representation a -> ο of the ring A
on U* (BU) is well-defined and is faithful.
848
S. P. NOVIKOV
Proof. The independence of the definition of a from the choice of representative (Χ, ζ) of the
class χ follows from the standard arguments verifying invariance with respect to cobordism of pairs
(Χ, ζ) and properties of Poincare-Atiyah duality for manifolds with boundary.
The fidelity of the representation a follows from the fact that it is already faithful on U* (P)
C i/# (BU) by the preceding lemma, where a coincides with ιφ'1 (a)]
to be verified i s that ο is a representation of the ring A
· The only thing that remains
and not of some extension of it. For this,
however, we note that the composition of transformations o i is also induced by some characteristic
class and hence has the form a b = c. Whence follows the lemma.
Remark 5.1. It is easy to show that the transformation a has the form φ~ια*φ,
where φ : U*(BU)
-» i/* (MU) and a*: U*(MU) -* U*(MU) is the transformation induced by a: MU ->MU. In the future we
r\_,
shall use the geometric meaning of the transformation a = φ' α*φ and hence we have given the definition of ο in a geometric form.
The transformation a induces a transformation α * : Ω;/ -> Ω(/ = V* (Ρ), where U\(P) = Ωί, = U'l(P).
We shall also denote by a* the dual transformation U* (P) -> U* (P), U* (P) = A = Qy.
We shall now indicate the set of operations needed, from which we can construct all the operations
of the Steenrod ring A .
1. Multiplication operators. For any element α € U* (Ρ) = Λ there is defined the multiplication
operator χ -> ax. Hence Λ C A . The corresponding transformation a : ί/*(βί/)->ί/*(βί/) has the form:
where (Υα, —τγ ) represents the element Da €.U*(P) = £ly.
2. Chern classes and their corresponding cohomology operations. As Conner and Floyd remarked
in L^M, if in the axioms for the ordinary Chern classes one replaces the fact that ci(^) for the canonical f/j-bundle over CP
is the homology class dual to CP ' l , by the fact that the "first Chern
class" σ1(ξ) is the canonical cobordism class ai € i / 2 ( C / ) A ) which is dual, by Atiyah, to
[CPN'1],
2l
then there arise classes σ^ζ) € U (X) with the following properties:
1. at = 0, { < 0; σ 0 = 1; σ £ = 0, i > dim c ξ;
2. σ, (ξ+η)=
2
0}(1)σκ(ΐ\)\
3. σ,(ξ) £ Map (X, MU) C U2{X), if ξ is a i/,-bundle;
4. ν (σι) = c j , where ν : U* -> H* ( , Z) is the map defined by the Thom class.
We note that in the usual way (by the symbolic generators of Wu) the characteristic classes ai
determine classes σω (ξ), ω = {klt ..., ks), such that σ ω ( | + η ) =
^j
Ρω^ξ) σ ω 2 ( η ) , ,
ω=(ωι, ω2)
with σ ( ι
ο = σι.
In the usual way the classes σω determine elements Sw = φσω € U* (MU) and, as was shown
earlier, homomorphisms σ*ω: Qy -» ily and Sa : (/* (BU) -» U*(BU).
We have the important
Lemma 5.4. The following commutation formula is valid:
849
ω=(ο>ι, ω2)
Proof. This formula can be established easily for the operation on t/* (BU) by the faithful representation which we constructed earlier.
Let (Χ, ξ) represent an element of U* (BU) and (Μ, —τΜ)
represent an element χ of Ω^. We consider
5 β οϊ(Χ > |) = 5 ω [(Χ,|)χ(Αί,-τ Μ )]=
2
ol(x)aa,(X,l)
ω=(ω,, ω2)
=
.2
(Y^fUl + tx)-^rjX(^2
-τ*ω)
ω=<ωι, ω2)
by definition. Here (Υωι, / ω ) represents the element Όσωί{ζ),
and similarly for /V^ . The lemma i s
proved.
In order that the formula derived above be more effective, we shall indicate exactly the action of
the operator σ% on the ring Qy.
It is known that by virtue of the Whitney formula the classes σω( — ζ) are linear forms in the
classes σω{ξ) with coefficients which are independent of ξ. Let σω{ξ) = σω(—ξ)
and let σω be the
homomorphism associated with this linear form.
If (Χ, —τχ) represents an element a of Qy, then the classes σ ω ( ο ) , represented by
(Όσω(—τχ)
μ ίwhere e is induced by X -> P\ are the characteristic classes of the tangent bundle.
Let X = CPn and ωω =
2
ti'· · • U
(the sum over all symmetrizations, ω = (klt . . . , ks)).
Let λ ω be the number of summands in the symmetrized monomial ιιω, k = ΣΑ^. We have the simple
Lemma 5.5. If X = [CPn],
then σ*ω(Χ) = λω [CPn-k] and
σ*(αδ)=
2'
σω,(α)σω 2 (6),
a, b, e Qv.
Hence the above formula completely determines the action of the operators σ*ω and σ*ω on the
ring Ων.
Proof. Since for X = [CPn] we have that τχ + 1 = (π + 1) ξ, where ξ is the canonical ί/,-bundle,
the Wu generators for τχ are u = i, = . . . = t n+1 = D C P " · ' € i / 2 ( Z ) .
Therefore σ ω * [ C P n ] = λ ω ϋ 4 ,
where k = dim ω.
We note that by virtue of the structure of the intersection ring 6 f *(CP") we have: u = DCPn~k.
Hence
Iwhere f : f* (CPn) -» ί/* (Ρ) is the augmentation!. The Leibnitz formula for a%{ab) follows in the
usual way from the Whitney formula.
The lemma is proved.
We shall now describe the structure of the ring S generated by the operators Ξω.
850
S. P. NOVIKOV
We consider the natural inclusions
CP?
and
x . . . x CP™-1* BUn-^MUn
homomorphisms
i': U
We note that U* (CP™ χ . . . χ CP™ ) has generators u{ € U2 (CP™ ), and an additive basis of
...
χ CP™) has the form ~LKqxqPg (ult. . . , un),
where xq € Λ = U* (P), the \
q
are integers and Ρq
are polynomials. We have the following facts:
1. The image Imt* consists of all sums of the form 1.XqxqP
q
(u,, . . . , u n ) , where Ρq is a sym-
metric polynomial and dim %{Pι = constant (the series is taken in the graded ring).
2. The image Im(i*/*) consists of the principal ideal in Im i* generated by the element ut . . .un.
3. The i*aq = aq (it,, . . . , un) ate the elementary symmetric polynomials, aq the characteristic
classes.
4. For any α € U*(BU„) we have the usual formula i*(a)(ul . . . un) = ί*]*φ{α), where φ is the
Thorn isomorphism.
From these facts easily follows
Lemma 5.6. The operations Ξω £ A
2
1 . If a € M a p (X, MUJ CU (X),
2 . So (αβ) = 2
then
have the following
S(k)a =a
k +1
properties:
and S w ( a ) = 0 if ω φ (k).
&>. (α) ^ω2 ( β ) for all α, β G U*.
ω=(α>ΐ, ω 2 )
3. Ifku)<n,
a i
= (k1(l\ ..., ks(p),
%k?
= ffl U σ = 2
λ
then
^ '
αφ(1)=αη = 0 is
equivalent to a = 0.
4. The composition
of operations
with integral coefficients,
SM -Sa
is a linear combination of operations
so that an additive basis for the ring S consists
of the form. Sa
of all δ ω .
Proof. Let X = BU\ = CP°°. Since MUl = CP™, it is sufficient to prove property 1 for the element
u GlJ2(CP™) equal to σι (ξ) for the canonical f/j-bundle ξ.
2
£ U (CP™) a n d S a ) ( I i ) = / * S a , 0 ( l ) = /*u'
for U1 -bundles ζ.
c+1
By definition, we have: it = ]*φ(1)
(if ω = (A)) and σω (ξ) = 0, if ω ^ (k), since σ, = 0, i > 2,
This proves property 1.
Property 2 follows obviously from the Whitney formula for the classes σω together with the remark
that φ{1) GU* (MUn) a s n ^ t » represents the universal element corresponding to the operation 1 £A
Property 3 is clear. Property 4 follows from the fact that on the basis of properties 1 and 2 it is
possible to compute completely S ^ · Sω
follow for large η that Ξω
The
(u) = Σ λ ω δ ω ( κ ) and then use property 3. Whence it will
o S ^ = Σλω5ω.
lemma is proved.
Further, we note the obvious circumstance: An additive topological basis of the ring A
l
form XiSu,, where %i is an additive homogeneous basis for U*(P), U (P) = Ω[/.
has the
.
The topology of A
851
is defined by a filtration. This means that the finite linear combinations of
1
the form XA^S^ are dense in A^ and the completion coincides with A , which thus consists of
formal series of the form ~2.λι%βω ., where the λ; are integers and άιταχβω -- constant, since A
is
a graded ring.
Thus we have:
where the sign / \ denotes completion. Here Λ = Ζ [xl, ..., x^ . . . J, d i m * ; = — 1i. The ring S i s
completely described by Lemma 5.6, and the commutation properties by Lemmas 5.4, 5.5.
We note that S is a Hopf ring with symmetric diagonal A:S^>S
Δ (5 ω ) =
Σ
<8> S, where
^ω, ® 5ω,·
(ωι, ω 2 ) = ω
Since MU is a multiplicative spectrum MU ® MU ^MU, the ring /ί
Δ:
where Δ ( 5 ω ) =
2
^->^
υ
®
Λ
4
υ
has a "diagonal"
,
^ω, ® Sa>, and «α Θ i = α Θ zi> = %(o ® 6) for « € fly = Λ. The Kiinneth
ω=(ω!, ω2)
formula for Ky, K2 GO {complexes without torsion! h a s t h e form:
U* (Κι X # 2 ) = C/* (
and hence /I
®A
is an .4 -module with respect to the diagonal Δ.
Moreover, we remark that A
h a s a natural representation * on the ring £ly, where Ω1., = U l(P),
under which the action of the ring Λ goes over to the multiplication operators Λ = Q,y and the SM-â* .
We now define an important map y : U2 -> U2 (nonadditive), such that νγ(χ) = v(x), u : U* ^H* ( ,Z)
is defined by the Thorn c l a s s , and γ(χ) € Map (A', ML',) C U2 (A), for χ €
U2(X).
We consider important examples of cohomology operations related to the c l a s s σι.
1. Let Δ ( £ ΐ ( jC2) C A
be the cohomology operation such that
σι) Λι ν(σι) Λ '] e
where σ, € U2(BU),
U'(MU),
y.U'Û3.
In particular, Δ ( 1 0 ) will be denoted by (3 and Δ(, t ) by Δ.
We shall describe the homomorphisms A* ^
k2) and Δ(^ , ^ ):
a) if (Α, ζ ) represents an element of f/* (BU) and i t : Yt -> X, i2:Y2 -> X are submanifolds which
realize the c l a s s e s Dcx {ξ), —Dcl (ξ) e Η n _ 2 (A), then their normal bundles in A are equal respectively
to ξ1 and ?!, where c t (f, ) = -c,^ (f,) = - c, ( | j = c, (f).
Let
852
S. P. NOVIKOV
be the self-intersection in U* (X) with normal bundle
** (^ + · ; · + .
ξι
w
+ ίι + y + . i i ) = >
kL
hi
where i: Ykl>kl -> X.
£)=jYikl. ft2), i * ( i + ^))·
WesetA(klfk2)(X,
b) If if= —'"ΛΊ then theA(fc
^ 2 ) define homomorphisms Δ (^^ j^).· Ωυ-»Ω(/ for which the image
of d consists only of S (/-manifolds.
The operations d and Δ on Ω;/ were studied earlier in [ 13 J.
2. The c l a s s e s and operations X(kli
k2)·
Just
2
classes γ{ο% ' ) y(— σ,) , the operations X(k
fined for a bundle ξ only a s functions of ο^ξ)
k
a
s w a s
t n e
case for the operations &(klt
k2) and
) and the c l a s s e s corresponding to them will be deor of γ(σι (ξ)).
We define these c l a s s e s for one-dimen-
sional bundles ζ over CPn.
We consider the projectivization Ρ(ξ + k)^CPn,
n
It i s obvious that τ{Ρ(ξ+ h)) = p* r(CP )
where k i s the trivial Α-plane bundle.
+ τ', where τ' consists of tangents to the fiber.
Over
Ρ (ζ + k) we have the following fibrations:
1) the Hopf fibration μ in each fiber;
2) The fibration ξ = ρ* ξ.
It is easy to see that the stable bundle r ' i s equivalent to the sum
k times
We set {here kl + k2 = k\
which functorially introduces a £/-structure into the bundle τ'(klt
) such that rr'(ki, fc2) = n where
kz
r i s the realification of a complex bundle.
Ρίξ+k)
has the induced t/-structure p* r(CPn)
(k
+ r\ki>
.
ki)
We denote the result by
n
+ k). We denote the pair (P >- ^ (ξ + k), p) € U*{CP ) by 0 χ ( Α ι ,
, where
ki)
X ( k l
Ρ^·^\ξ
, k2) € U* (CPn).
For any fibration ξ over X we set X(kl,ki) (ξ) = X(klt k2) (fi), where cl (ξ) = c, (^) and ^ is a
i/, -bundle.
There arise classes X(klr k2) e ^ * (BU), operations φχα^, k2) = ^(klr k2),
a n d
homomorphisms
We note that χ(ο t ) = 0. We denote the operation X(l 0 ) by χ and the operation χ(ι χ ) by Ψ.
The homomorphism Ψ*: Ω^/^Ω^ was studied by Conner and Floyd (see t 1 ? ] ) .
It i s easy to establish the following equations:
a) A(ki kl)°d
= 0(in particular, d2 = 0, Δ<9 = 0);
b) ΔΨ= 1, [d, χ] = 2, Xd = Xlod,
where ^ = [CP1]
€ Λ C A U; ΘΨ = 0.
We shall prove these equations. Since Im<9* C Ωμ i s represented by S(/-manifolds, Δ (ki> ki
= 0 by definition; since * i s a faithful representation of the ring A by virtue of Lemma 3.1,
^•(.ki, k2) °d = 0, where d = Δ ( ι > 0 ) .
853
The equations Δ* Ψ* = 1, θ*Ψ* = 0 were proved by direct calculation in L ^ J . Hence Δ Ψ = 1
in A . Since Im<5* consists of SU -manifolds, it is easy to see that χ* d* = χ, ο d*. This means that
χθ = χβ. The equation [χ, d] = 2 follows easily from the fact that for one-dimensional bundles ζ
over X such that c t (.f) = — c^X), we have:
U
and the class DC is realized by the submanifold X = Ρ(ξ) C Ρ(ξ + 1).
Remark 5.2. Equations of the type ία, b] = λ ο 1 arise frequently in the ring A . For example,
if α£ = S/,. and b^ = [CPk], then [a^, bk] = (k + 1) ο 1 by Lemma 5.5.
Remark 5.3. The operation η = [Δ, Ψ] = 1—ΨΔ is the "projector of Conner-Floyd" π2 = π. (Conner
and Floyd studied π*.)
This projector has the property that it allows the complete decomposition of the cohomology
theory U* into a sum of theories nfU*, where Σπ ; = 1, 77, € A U, with n0 = 1 -ΨΔ and 77, = ψ>Δ' -Ψ>'+1
Δ ; + ι . Later on we shall meet other projectors of this same sort.
3. We consider still another important example of a cohomology operation in (7*-theory, connected
with the following question:
Let ζ, η be (/-bundles. How does one compute the class σ^ζ
® η)?
We have
Lemma 5.7. a) For any Un-bundle ζ there is a cohomology operation y n ., € A
σ
1
= Yn-ι ( η ( Ό ) > where λ_, = Σ( — 1)'Λ' and the Λ are the exterior
such that σλ(
powers.
2
b) If ul, . . . , un G U (X) are elements in the subset Imy = Map (X, Ml)\) C U2(X), then we have the
equation
Yn_l(lii
. . . Un)
= Υι(«1·γΐ(Μ2·γι(...·νι(κη_ι·ΙΙη)) · · · ) .
where y, is such that for a pair of Ui-bundles ζ, η we have the formula
The proof of this lemma follows from the definition of the operation y t . Let X = CP°° χ CP°° and
let ζ, η be the canonical ί/,-bundles over the factors. Since νσ^ξ ® η) = c^t; (g) 77) = c^tj) + £^(77)
and CTt € Map (,Υ, Λ/ί/χ) it is possible to calculate the class σι (ξ ® 77) completely as a function of
at (ξ) and σ, (77). Namely:
Since the bundle λ, (ζ + η) lies in a natural way in K" (MU2) and λ! (ζ + η) = ζ ® η — ζ — η + 1, the
difference - σ , (f) -σ, (77) -KJ-J (£ ® τ;) + 1 has the form γ^, where u G U* (MU2) is the fundamental
class u = <y!>(l).
The operation y, can be written in the form
Yjii = 2
a;j,Ai,j)(«i"2),
Μ =
854
S. P. NOVIKOV
where u t = σ, (ξ), u2 = σί (η).
Let ω = (kt, . .. , ks), where s > 2. Then SM(u) = 0. Hence yt is uniquely defined (mod χω$α>)·
We set y n ., = y1(u1 . . . y ^ u ^ O . . .) on the element u = ul . . . un = 0(1) € U* (MU„). The operation γη.ί is well defined υα.οάχω8ω, where ω = (klt . . . , A s ), s > n. By definition, we have the formula
σ,λ.! ( f ) = γίι.ι σ η (^) for a £/n-bundle f.
Remark 5.4. It would be very useful, if it were possible, to define exactly an operation γί € A
® (? so as to satisfy the equations y t ' = γ';. The meaning of this will be clarified later in § 8 .
We now consider analogues of the Adams operations and the Chern character in the theoru of Ucobordism which are important for our purposes.
We have already considered above how the class σ, (ζ ® η) is related to the classes σχ (ζ) and
σ1 (η ) for ί/,-bundles ζ, η. Namely
σι(1®η)= u + ν+yi(uv),
where u = σι (ξ), υ = a t (η) and
ίφ}
We set u + ν + γι (u, v) = f (u, v). Then we have the "law of composition" u Θ υ = f (u, v) for u, ν
€Imy t = Map(/¥, MU^), which turns Map(î, Mil\) into a formal one-dimensional commutative group with
coefficients in the graded ring Oy, while dimu, v, f(u, v) = 2. As A. S. Misc'enko has shown, if we
make the change of variables with rational coefficients
where [CPl] € Ω^| = Λ" 2 1 , then the composition law becomes additive:
g(u®v)= g(f(u,v))=g(u)+g(v)
(see Appendix 1). This allows the introduction of the "Chern character"
a) We set σΗξ) = egU\
where u = σ, (ξ) for ί/,-bundles ξ;
b) if ξ= ξί+ ξ2, then σΗξ) = σΗξ) = σλ(£) + σΚ{ξ2);
c) if ^ = ^ ® 4 , then σΑ(^) = σΗξ^
Thus, we have a ring homomorphism
ah:
We now consider an operation a € A
such that
We already know some examples of such operations:
1) a =
2)
a=
855
2-^(1 Di»0
. '
i times
The Chern character gives a new example of such an operation a € A 6$ Q: We consider the
"Riemann-Roch" transformation λί"*: U2n->K which is defined by the element λί""1 € K(MUn), and
let λ = (λί;°), η - oo. Let Φ ( π ) = σ/ί" ° λ ^ ) , and Φ = (Φ(η)).
The operation Φ obviously has the
property that Δ Φ = Φ ® Φ since ah and A.t are multiplicative, and if the element ^C/iiA 7 ) has filtration m and the element η has filtration ra, then ahm + n (ξ ® η) = σhm(ξ)σhn(η).
It is easy to
verify that the operator Φ has the following properties:
1) Φ2 = Φ,
2) Φ*(1)= 1,
3) Φ* (Λ;) = 0, dim» < 0 , Ϊ 6 Λ , Λ = U* (Ρ), Φ*: Λ - Λ, where
5
2
" ^
>'
Hence, the operation Φ associated with the Chern character ah defines a projection operator, which
selects in the theory U* <g> Q the theory //*( , Q) = Φ (ί/* Θ Q).
A multiplicative operation a G AU is uniquely defined, obviously, by its value a(u) £ U* (CP°°),
where u € Map {CP°°, MUt) is the canonical generator, a(u) = u (1 + . . . ) .
Conversely, the element a(u) € U* (CP°°) can be chosen completely arbitrarily. For example, for
; for ah =
a = "y. So,, a(u) =
^
ί —u
3^! 5 ω , a(u) = u (1 + uk).
ί Λ
-
«
o>=(ft
h)
For our subsequent purposes the following operations will be important:
1) The analogues of Adams operations Ψ^ £ AU <g> ZZ [ l / p ] .
2) Projection operators which preserve the multiplicative structure.
All these operations are given by series a(u) € U* (CP ), since Δα = α ® a.
We define the Adams operations Ψ^ , which arise from the requirements:
i)
Ψ(χν)Ψ
2)
Ψυ • χ = tex • ψ £ . r«e χ e Λ " 2 ί —
3)
Ψ[/(Μ)
fc
Μ® . . . θ α
= —
Ω";
(A times), where u € U2(CP ) is the canonical element and
k
composition in Map(,¥, Λ/ί/χ) C U2(X).
Ik
Lemma 5.8. a) The series xH.,(u)has
the form
-υ
1
= —f(u,f(u,
x
b)
Ψ* * (x) = kix,
'
k
A"
/<•
χ e A~zi == υ-Μ(Ρ) =
...J
S. P. NOVIKOV
856
j\
a>
XI) k ψ I __ m kl __ xu I m k .
U U ~ U ~ UΎ U '
e) for a prime p, all λ;, λ; G A'21, such that
are integral for i < p. Hence the element ρπΨ£ (ut . . . un), where ux . . . un € U2n(MUn) is a universal
element, is integral, and so the operation ρ"Ψ£ for elements of dimension 2n is "integral," if the
dimension of the complex is < 2pn. (See Appendix 2 for proof.)
4. We now consider the projection operators. The condition defining a projection operator
π € A is obviously π2 = π, or η*2 = π*, where π* : Λ ^ Λ is the natural representation. We shall consider only those π for which π(χγ) = π (χ) n(y) and π{ιι) = ^XiU1 e U* (CP°°). Let
Xi = [cpq,
JI(U) = (ι + 2 *,-»*)«.
>1
where the λ ; € Λ ® Q are polynomials in xt with rational coefficients, dim λ ; = — 1i. It is easy to
show that π *(λ ; ) = 0, since π2 = π.
We shall be especially interested in the case when there exists a complete system of orthogonal
projectors (ττ;-), ττ^π^ = 0, / ^ λ, which split the cohomology theory U* into a direct sum of identical
theories.
Let y € Λ and Δ χ C Λy ® (? be the "operator of division by y," which has the following properties:
1) Ay(aft) = Ay(a)b + aAy(b) -
yAy(a)Ay(b),
2) A y *(y)= l .
L e t O y =yAy, Ψχ = 1 - Φ χ € AU <8> (?. It is easy to see that Φ^ = O y , Ψχ =Ψ y ,andΦ y oΨ y = 0.
Moreover, the collection of projectors π £ = JlAly — yl +1 is such that πρ^ = 0, / Φ k, and it decomposes
the theory U* ® Q into a sum of identical theories.
Let y] e A ; = Ω^; be a system of polynomial generators, and Φ£ = yiAy .. We note that Φ* (y ; )
= 0 for / < i. Let ^ = yk for A < / and y 4 = (1-0ί)*7Α =ψί* (ΧΑ)· Obviously, Φ £ * ( η ) = 0 for k φ i
and Φ*(γί) = yi = y'i.
Since (1—Φί)*(χ;·) = y ; for / < . i and yj—yiAyi(yj)
yk is a system of polynomial generators.
= (1—4>i)*yj for / > i, the collection of elements
The projectors nf = y/&' -yi' +1 Ay.' +l clearly are such that 7 7 * / : Λ ^ Λ carries monomials of
the form y .'γ\ , .. ., γ ι , j > 0, into themselves for t,, .. . , is Φ i, and all other monomials into zero.
This means that
and
Kerjt/= U
In particular, 1— Σττ;· and ττ;+1 = y^jAy..
/
iTj (U ® ζ*) are isomorphic.
Hence
= ?r;-(y^) for all % e ί/ , and all theories
857
The projector na = 1 —y,A y [ h a s the following properties:
a) πο{χγ) = no(x)no{y),
i.e., π0 is multiplicative.
b) The cohomology ring of a point for the theory TTO(U* <8> Q) has the form Q(y:, . . . , ) ; , . . . ) , where
ô*(y ; ) =Yj f o r ; 7^ t .
c) All theories TTS(U* ® Q) are canonically isomorphic to the theory no(U* <S> Q) by means of the
operator of multiplication by y s , and their defining spectra differ only by suspension.
Examples of operators A y : if dimy = 2k, i.e., y G Ω[/2 = U'2k(P),
and ff*ft>y = - λ ^ 0, then we
set
y=
For the generators Ji €
2 J
we have | λ | = 1 for i ^ p' — l and | λ | = ρ for i = p ' — 1 for any prime p .
Hence
v{ = Σ
yiq~isa,...,i
and
q-1
It i s easy to s e e that for i + 1 Φ ρ' for given p , A y . € /I
€ /I
® Z*?p> i ° r i + 1 = p ; and ρ > 2, Δ
<E> (^, where Qp is the p-adic integers.
Now let y ι be a collection of polynomial generators of Ω^ and let ρ be prime. We consider a l l
numbers i j. ^ p ; — 1 in the natural order, i\ < t 2 < . . . < ij. < . . . . Let Φι = (1— y, Δ
), where k is
^
* ' lk
some sufficiently large integer.
The projector Φ^ is such that the ring Φ^ Λ C Λ has a s a system of
polynomial generators all y\ for i Φ i^, and Φ^. yik = 0.
Obviously, the operator Φ^ commutes with the operator of multiplication by y ; for / < t/c, since
φ
ί: = 1 -Jik^Yi,'
a n d
^y,, commutes with yn j < ih.
We consider the operator Φ Λ .Δ χ
Φ Λ = Δ^.,.
Since Φ^ i s multiplicative, Δ ^ i s the operator of
division by Φ^ί^.,Φ^. Hence in the cohomology theory Φ^.(ί/*) the operator Δ^.., h a s a l l the proper'V/
"^.
. ^.
t i e s m a k i n g 1 — y i h a t^lc.i = ^k-ι a m u l t i p l i c a t i v e p r o j e c t o r , a n d Φ , / ,= γ'. Δ ; — y ' +1 Λ '+1 f o r m s a
ft*i
•
complete
projectors.
T h u s ,system of orthogonal
Φ Λ = Φ Λ ( 1 - yI
-y
/£i
l f c .,
h e r e Φ λ = 1 - y i A . A y .^. If <t>s = 1 _ y ^ A y .^, t h e n w e s e t :
where
^, or:
^ - ^ ' ^ ' ^ h ' l k - i
h and
A; "i
= Φ4-ι, while
858
S. P. NOVIKOV
φ™,, = φ™ φ Η φ ™ ,..., φ}" = Φ ^ Φ , Φ ^ ..., ΦΙΜ = φ,1*1
[ft.]
The projector Φ
is obviously such that
ys, s + 1 = P> for
S s=; ift,
υ
b) φΐ*] e= ^4 ® ζ Qp.
The collection of Φ'
and hence the sequence Φ
Φ € /I
with A -> °° is such that Φ * is independent of k when it operates on Ω^
as 4 -» », or the series 2
(φ^+'Ι — φ Μ ) = φ
defines a projector
® zvp which is multiplicative and such that:
a)
O*(ys) =
b) Φ2 = Φ,
c) the theory U* <8> %QP splits into a sum of identical theories of the form Φ(ί/* ® z@P) U P t o
shift of grading (suspension).
a
We note that the elements γs = Φ*(γ!.) for s = p' — l have the property that all σ£ ( y s ) ^ 0 modp
for all ω, dim ω = 2s.
The cohomology theory Φ(ί/* ® zQp) is given by a spectrum <W(P), where Η (Λ/(ρ), Ζρ) = Α/β Α
+ Αβ, A the Steenrod algebra and β the Bokstefn homomorphism.
Thus, we have shown
Lemma 5.9. a) There exists a multiplicative projector Φ € A ® z(? p sucA ίΛαί ί/ie cohomology
theory Φ(ί/ ® zQp) ' s given by a spectrum Λί(ρ), where Η (Μ(ρ), Ζρ) = Α/Αβ + βΑ, and the homomorphism Φ : Λ -> Λ annihilates all polynomial generators of the ring A = U (Ρ) = Ω(/ of dimension
different from p' — l.
tAeir
b) Γλβ theory U <£) zQp decomposes into a direct sum of theories of the form Μ
suspensions.
(p
) = [/
p
arao?
§6. The /4 -modules of cohomology of the most important spaces
In this section we shall give the structure of the module U* (X) for the most important spectra
X = Ρ (a point), A: = CPn, X = ffP2", X = RP 2 "· 1 , Ζ = MSU, X = S2n-VZp, X= BG, G = Zp.
1. Let A7 = P. The /4'"'-module ί/* (Ρ) is given by one generator u € U" (P) and the relations Su(u)
= 0 for all ω > 0. An additive basis for U*(P) is given by the fact that U*(P) is a free one-dimensional
Λ-module, where Λ = Ω^. We shall denote the module U* (P) by Λ.
Clearly, we have:
HomA* υ (A", A) = U. (P) = Qv.
If d: A -> A is a map such that d(l) = a € A , then it is easy to see that d*(hx) = ha*^x-}, where
hx gHom^ y(A U, Α), χ G A, and hx is such that hx{\) = x.
859
In particular, for a = S^ we have α* = σ*ω , and for a = d, Δ we have a* = d* or Δ*, the known
homomorphisms of the ring Ω[/.
These remarks are essential for computing
ExtAu(
2. Let X = CPa = (EkCPn)
, IT(P))=
ExtAu(
,Λ).
€ S. It is easy to see that U*(X) is a cyclic module with generator
u € t/2(Z), satisfying the relations:
a) Sw(u) = 0, ω ^ (k),
b) S(k)(u)
= 0, k > n.
These results are easily derived from the properties of the ring U*(CP") and the properties of the
operations δ ω given in Lemma 5.6.
3. Xlk»> = S2n + l/Zk
=(EkS2rl+VZk)eS,
X2in) = RP2n
+i
. U*(X) has two generators u d
U\X{kn)),
υ G U2n+1 (X k), satisfying the relations:
a) Ξω(ιι) = 0, ω £ (q),
b) S(q)(u)
= 0, q>n,
c) ( Α Ψ ^ Χ » ) = 0, u € Map a
t j
Mf/,),
d) Su(v) = 0, ω > 0.
These results follow from [7] for K* (BG), G = Z p ) and the σ, K° ^ t/2 and the ring U*(BG).
4. For X = # P 2 n , SG, the module U* (X) is described as follows:
a) U*(RP2n) = U*(RP2n + l)/v.
b) U*(BZk) = lim [ £ / * ( ^ n ) ) ] .
5. We now consider the case Λ7 = MSU. Since t/* (HiSi/) = φϋ* (BSU) and St/-bundles are distinguished by the condition cl = 0, which is equivalent to the condition γσι = 0, we have U* (MSU)
= U* {Μυ)/φΙ{γσι),
where / is the ideal spanned by ( y a j , / C U* (Si/).
The natural map U* (MU) -> U* (MSU) is an epimorphism. Hence U* (MSU) is a cyclic Λ ^-module
with generator u € U" (MSU) and au = 0 if and only if α €
In particular, au = 0 for a = Δ(^
φ](γσι).
j. ).
We have the important
Theorem 6.1. a) 77ie module U*(MSU) is completely described by the relations d (u) = 0, Δ(κ) = 0.
b) The left annihilator of the operation d consists
of all operations of the form ad + έΔ, a, b
v
eA .
Proof. We consider the module Ν = Αυ/ΑυΔ,
+ Aud and the natural map f:N -> U* (MSU). We
shall show that this map is an isomorphism. Since for the operation Δ there exists a right inverse
Ψ such that ΔΨ = 1 and (9Ψ = 0, the module Α Δ is free, and it is not possible to have a relation of
the form οΔ + bd = 0 if α φ 0 or bd φ 0.
We now consider A d.
We shall establish the following facts:
1) The left annihilator of the operation d consists precisely of the operations of the form
φΙ(γσι)€Αυ.
2) The operations of the form A d form a direct summand in the free abelian group of operations
A
under addition.
860
S. P. NOVIKOV
We consider the representation α - > ο Ό η U*(BU). Let ζ be an SU-bundle.
It is easy to see that
we have the equation
where cl (η) is the basic element of H2(CP1).
It is also obvious that Im<9 consists only of pairs (Χ, ζ)
&U*(BU), where ξ is an 5t/-bundle. Hence Imd is precisely U*(BSU).
Whence follows fact (1).
For the proof of (2) we note that U*(BSU). is a direct summand in U*(BU). We decompose U*(BU)
into a direct sum U*(BU) = U*(BSU) + Κγσ,).
Then U*(MU) = A U decomposes into a direct sum A + B,
where Β is the annihilator of U*(BSU) with respect to the representation "a. Obviously, A <3 = (S-f A)d
= Ad. If the operation a fiA is such that ad is divisible by the integer λ, then ad is divisible by λ,
and hence for all Si7-bundles ζ the characteristic class φ 1 α(ζ) is divisible by λ. Hence this class is
a λ-multiple class in U*(BSU) and (up to JiyaJ)
a λ-multiple class in U*(BU). Whence follows fact
(2).
We deduce from (1) and (2) that the map /: Ν -» U*(MSU) is a monomorphism. Since Ν = AU/AU
A
+ A d, it follows from (1) and (2) that the kernel Ker/ is a direct summand. Since Α Δ is a free module and A d is a module isomorphic to U*(MSU) with shifted dimension (see (1)), the equation Ker /
= 0 follows from the calculation of ranks in the groups
(AUA Θ Λ Ζ)h = #<h-4> (MU, Z), (Aud ® A Z)k = Bkl (MSU, Ζ),
(U'(MSU)® Z)k = Hh(MSU,Z).
Thus, U*(MSU) = Αυ/ΑυΔ
+ AUd. Since the left annihilator of the operation d is precisely the left
annihilator of the element u € U° (MSU), it follows by what was proved for U*(MSU) that this left annihilator is precisely Α Δ + Ad.
The theorem is proved.
§7. Calculation of the Adams spectral sequence for U*(MSU)
In this section we shall compute the ring
Ext*Au(U*(MSU),A)
and all differentials dt of the Adams spectral sequence (Er, dr), where E2 = Ext
(U (MSU), A). In
(J
particular, it turns out that d{ = 0 for i ^ 3, d3 Φ 0, and E^ = E\' = 0 for i > 3For the calculation of Ext*j \j we consider the complex of A -modules
e
C=
d
d
{U*{MSU)+-C0+-Cl*-...<-Ci...),
where the generators are denoted by ut € Ct for i > 0 and vt € C ; for i > 1, Co = A
for i > 1. We set rf(u£) = du^ and d(vt) = Au^.
and C ,· = A + A
Since d2 = 0 and Ad = 0, d2 = 0. It follows from the
theorem above that C is an acyclic resolution of the module U*(MSU) = H0(C).
We now consider the complex Horn*Au(C> Λ), where Λ = U*(P). Since Horn*AU(AU,
obtain the complex
d*
HomJXC, Λ) =
where d* = d* + Δ*:Ω[/^Ωί/ + ily.
d*
d*
A) = Ων, we
Since Δ* is an epimorphism, the complex Horn* Ay(C,
8·
β*
861
Λ) reduces to the following:
a*
where W = KerA* Cfly.
From this we deduce the following assertion.
Lemma 7.1. a) For all s > 1, we have isomorphisms
X
b)
(£/'(MSU),A) = Ht-2s(W, d').
ExtlV (IT (MSU), A) = Ker d* Π Ker Δ* c
Ών.
c) If h £ Ext^'li (U* (MSU), Λ) = Z2 is the nonzero element, then the homomorphism α -> ha: Ext ι'υ
-> Ext t+.}'
is an epimorphism with kernel Imd* for i = 0, and an isomorphism for i > 1 (we recall that
the spectrum MSU is
multiplicative).
Proof. Statements (a), (b) of the lemma obviously follow from the structure of the complex W, in
which the grading of each term is shifted by 2 from the one before by the construction. For the proof
of (c), we note that h = i^ d* O J , where xt = [CP1] € Sly, xt C W, and d*(x1) = — 2. Further, we note
that d*(xiy) = — 2y if d*(y) = 0. Hence the element hy is represented by the element -7rd*(yxi) for a
representative of y € / / * (W, d*). But since -K-(3*(y«1) = γ under the condition d*y - 0, statement (c)
Li
is proved, and therewith the lemma.
We consider the element Κ = 9x1 - 8x2 € Ω4^, where xx = [ C P 1 ] , x2 = [CP2]. Clearly, d*K
= Δ X = 0. The element Κ is a generator of the group
ΕΧΙΛυ [U* (MSU), Λ) = Ker 5* f] Ker Λ* = Z.
Since A IK\ = + 1, where /4 = e Cy2T and Γ is the Todd genus, by virtue of the Riemann-Roch theorem
there is an i such that dt(K) ^ 0 in the Adams spectral sequence, since for all 4-dimensional SUmanifolds the A -genus is even (see LÔJ).
It follows from dimensional considerations that d2(K) = 0 and d^(K) = A3.
We note that from dimensional considerations it follows trivially that d2/c = 0 (see theorem in §2).
Consider the differential
"3 ·
where d^K) = h\
V
&3
—>~ίί3
,
We have
Lemma 7.2. // a. € Εξ·
=F
q
for ρ > 3, and d3 (a) = 0, then a = di (β). Hence £ / ·
q
= 0 for ρ > 3, and
Proof. Let ί/3 (α) = 0; since a = Λ3β from Lemma 7.1, d3(a) = rf3 (Λ3β) = 0. Hence d3 (β) = 0
since multiplication by h : £ 3 P " -» £ 3 P + 1 · is a monomorphism for ρ > 0. This means that α = άι(Κβ).
Since
Σ £3P'* = Σ ExtjV (£
862
S. P. NOVIKOV
is the ideal generated by the element A, we have Εξ' q = 0 for ρ > 3From dimensional considerations it follows that ΕΛ
0
Since E*J = £ ^* + Eli* + Ell*
2
= A£oo' = AZ?So
i s
= E^ .
associated with Qsu
= π, (MSU), and E^* = hEaJ\
Eli*
we obtain
Corollary 7.1. a) flj*
+1
= AOj* ; b) Α2Ω2£, = ΤΟΓΩ 2 /^ 2 .
The equality (a) was first established in [ 1 8 ] by other methods, and(b)in [ 1 2 L
Corollary 7.2. a) 7*Ae image ofΩςυ ^n Ω;/ l
iy setting
equal to zero a certain collection
s
singled out within the intersection
(Ker d* f] Ker Δ * ) * = Es 2h-+
d3:
Kerd* f] kerA*
of linear forms mod 2, generated by the homomorphism
3
2fe 4
Εξ12h+2
Ά
= A (Ker 5* Π KerA*) - = Η ^ ~ * = 2
b) ΓΑβ group Ext 1 ^ = ff(IF, (?*) is isomorphic to the direct sum Ως^ + Ωο^Τ5, orerf iAis isomorphism comes from the differential d3:.
where Ext 1 ' 2 ^ 1 = Z?^'2'' = ^ J ' 2 * 3 ^ ^ J 1 = Kerrf3,h'3d} is well-defined since h3 is a monomorphism on
E x t ^ " 4 , and the image lmh'3d, = Ker d3 = Ω2,^5 C E x t ^ " 4 .
Corollary 7.2 follows from Lemma 7.2.
Remark 7.1. Part (b) of the corollary explains the meaning of the "Conner-Floyd exact sequence"
(see [13])
since H2k_2 (W, <?*) = Extl${U*
(MSU), A).
We note now that the groups H*(W, d*) ace computed in [ J 3 ] : namely, Hik{W) = Hski4(W)
= Z2 +
. . .+Z 2 (the number of summands is equal to the number of partitions of the integer k), Η ^W) = 0,
i φ 8k, 8k + 4. Whence we have:
ΕκΙ^2 (U' (MSU), A) = Exti' h + 6 (IP (MSU), Λ)
and
(U*(MSU),A) = 0, i¥=8k + 2,
8k + 6.
We have
Lemma 7.3. a) E x t 1 ' , 8 * 4 6 = Κ E x t 1 · ' * * 2 .
w h e r e
b) d3 (Ext°'j) = 0 for i ^ 8k+ 4, and d3 (Ext°''uk
K
+
€
E x t 0 · ? , ^ * (MSU), A).
" ) = Ext 3-'(Jh + 2 is defined by the condition
d3(K) = A3.
Proof. Suppose both parts of the lemma proved for k < k0 — 1. We show that d3(Ext°'y
fact, by the induction hypothesis on the groups Ext 3 '?,
°) = 0. In
the differential d3 is a monomorphism.
863
Hence E x t 0 · ' * 0 ^ 0.
A
u
We now consider d,(K Ext"· 8 ^ 0 ) = /i'Ext 0 ' 8 ^ 0 . We see that d3(K Ext 0 ' 8 ^ 0 ) is an epimorphism onto
Ext
3,eJc 0 + 6_ Whence parts (a) and (b) of the lemma follow; on the group Ext 3>8 ' c + 6 the differential is
trivial, and on the group Ext 3 > 8 f e + 2 D Ket d, = 0.
Thus, we obtain
Corollary 7.3. a) The image Ω,ί../Tor COy coincides
8 ί +4
b) For i = 8k + 4 the image Ω ^ ^ / Τ Ο Γ C Ω[/
"Riemann-Roch
with Ker d* f| KerA* for i £ 84 + 4.
is picked out precisely
by the requirement of the
Theorem":
where X is an SU-manifold, ζ £ kO (X).
We note that (a) follows immediately from the lemma. As to (b), we note that A [K] = 1. In [ 9 ] ,
"Pontrjagin c l a s s e s " π^ € kO* [X\ are introduced in AO-theory.
Consider the classes π2ι 6 kO (X);
[X] for Χ €Ω 8 ^/ΤΟΓ C fl'y. These num-
let TT2I = /<;. Now consider the numbers ch(cKti . . . cKik)A(X)
8
1
bers are different from zero mod 2 if and only if hX Φ 0 in Ω ^ .
Hence the condition d3(KX) = hlX^
0
in E\'* is equivalent to the fact that one of the numbers chicix^, . . . , Kik)) A(X) [Χ] φ 0(2). All such
numbers are in 1-1 correspondence with partitions of 8k into summands (8/,, . . . , 8/4.) (these facts are
easily deduced from [9]).
Since c h ( c / < v . . cKlk
<g) I) A {Χ χ Κ) [Χ χ Κ] = ch(cK^ . . . CKlh)A(X)
[Χ] °/ί [Κ], Α [Κ] = 1,
we have found elements κ^ . . . κ ^ <8> 1 € kO(X χ Κ) which do not satisfy the Riemann-Roch theorem,
and they determine n(k) linearly independent forms mod 2, where n{k) is the number of partitions of
k.
From this part (b) of the corollary follows.
The results of the lemmas and corollaries of this section together completely describe the Adams
spectral sequence for U* (MSU).
*^8. /c-theory in the category of complexes without torsion
Here we shall consider the cohomology theory k*, defined by the spectrum k = (kn), where
2η
= 0, i < n, and Ω 1ί2η = BU χ Ζ.
n
i(kn)
The spectrum k is such that the cohomology module H*(k, Z 2 ) is a
cyclic module over the Steenrod algebra, with a generator u € H"(k, Z2), satisfying the relations
Sql(u) - Sqz(u) = 0. Hence the spectrum k does not lie in the category D of complexes without torsion.
There is defined the "Bott operator" χ :k2n
and k2n is a connective fiber of BU.
-> k2n_2 by virtue of the Bott periodicity £t2k2n = k2n.2,
Since k°(X) = K°(X), we have on k° the Adams operations (see
[2])
defined by morphisms Ψ :BU -> BU such that Ψ* : n2n(BU) -> nln{BU)
is the operator of multiplication
by the integer kn (see [ 2 J concerning the operation of Ψ onK° (S2n) = n2n(BU)).
By virtue of this, the
operators Ψ can be extended to the whole theory Κ* ® Q, starting from the identity
864
S. P. NOVIKOV
where χ :JK'-> K1'2 is Bott periodicity.
In the category D of complexes without torsion the operator xn:k2n(X)
-> k°(X) is such that its
image consists precisely of all elements in h°(X) = K°(X) whose filtration is > In; moreover, χ is a
monomorphism.
In the category D we define an operation (&"Ψ ) by setting
where (knX¥k):
kln(X)
-> k2n(X).
It is easy to see that this is well defined and gives rise to an unstable operation (&ηΨ ) such
that (Α"Ψ ) can be considered as a map k2n -> k2n for which (knXVk)if:
n +
plication by k
T2n+2j(k2n)
-» "i n + 2 7(^2n ) * s multi-
>.
Let α η : = = ^ λ ^
(Α;ηΨΛ),
where the A^
are integers, be an unstable cohomology operation
ft
and a^*
multiplication by ^j λ&
kn+K
h
Definition 8.1. The sequence a = (an) will be called a stable operation if for any / there is a
number η such that for all Ν > η the number dy =
^jkh
kN^
is independent of Ν.
h
Definition 8.2. If the stable operation a has a zero of order q in the sense that a*
q
i+2q
then we also call b = (x'"a) a stable operation, where a = x b, b :k\X)^k
(X),
;
= 0 for / < q,
X €D.
We consider the ring generated by the operations so constructed and the operation χ by means of
composition, taking into account the facts that kx^k
= }¥kx and ψ^ψ
=ψ
. The resulting uniquely
defined ring, which we denote by / 1 ψ , is a ring of operations acting in the category D. In it lies the
subring of operations generated by the operations indicated in Definition 8.1 together with the periodicity x. This ring we denote by Bk
C Α ψ . There is defined the inclusion Βψ
-> Αψ.
We shall exhibit a basis for the ring Α ψ . It is easy to see that it is possible to construct operations δ; G Αψ
of dimension 2t, where δ0 = 1, such that the elements χ δ ; give an additive topological
basis for the ring Αψ , and all elements of Αψ
can be described as formal series SA^x^g^
where
the Aj. are integers. The choice of such elements δ; is of course unique only mod xA ψ (elements
of higher filtration).
We construct these elements δ ; in a canonical fashion: it suffices to define operations y; = Χ;δ ;
of dimension 0. Let δ 0 = 1. Let y/ί* = 0 and y/*^ be multiplication by 2. By definition, we shall take
Ύ iJ
= 0 for j < i and y.*
Yi = _/j μ^ kn+i,
to be multiplication by a number γ ι which is a linear combination
where the numbers μ^
are such that
/^j μ& kn+i
= 0 for / < i. We require in addi-
tion that γ ι be the smallest positive integer of all linear combinations of the form 2j l-1^
k
under the conditions:
2
h
μΛ1>Αη+ί = 0,
} < i.
kn+i
u
n
k
We consider the operation din — 2 . ft (k W ).
865
Here re is very large compared with i. It is
easy to see that the number γ ι does not depend on re for large re -> oo. Hence the operation is welldefined.
Consider the operations ain forre-> °o; we shall successively construct the δ ; from them. We have
aon = 1; let bmn = aln + κια2η + . . . +Kmamn
(fcmn)«
be linear combinations such that the homomorphisms
for / < m <3i re are multiplications by integers y;,/, where 0 < y 1 > ; < y ; . Clearly the numbers
y 1 > ; are uniquely defined. Let m -> °°,re-. oo; then in the limit, the sequence (bmn) gives an operation
ι)
which we denote by y t = Λ^δ,. It is uniquely defined by the properties that y, * = 0, Υι* = 2, and
in
o<Yl. <Yi,
yJ^n,,.
The operations γ ι are constructed in a similar fashion, and are uniquely determined by the conditions Yiin = 0, / < i, y#> = Yh and 0 < γ ^ < yk for k > i.
We exhibit a table of the integers y; * < ; ) = y^ in low dimensions:
0
2
4
To
Ti
T2
1
1
1
0
2
0
0 . . .
0 . . .
24 . . .
By definition, δ; = χ'1 γ ι. It is clear that the operations γ ι commute. Since π2ί(Βϋ) is Z, the rings
Α γ and β^,. are represented as operators on k*(P) = Ζ ίχ] in a natural way, in particular, the operations of dimension 0 by diagonal operators with integral characteristic values; the operation χ is
represented by the translation operator (or multiplication by χ in k*(P)). It is easy to show that we
have a transformation * : A^.^> Αψ such that *(Βψ ) C β ψ and ax = xa*. This transformation * is
completely determined by the condition that in k* ® (^-theory we have kx}Vk = x¥kx and *ΨΑ: = k*Vk.
We also indicate the following simple fact.
Lemma 8.1. The greatest
common divisor
of the integers
= (xl8i)lq^
for all i > 0, for a
common divisor of the numbers k (kq — 1) for all
fixed integer q, coincides
precisely
k.
a ^ j n G Β „, such that a^,,* = kn + > for j < /(re), where /(re) -» t» as re -» oo.
There exist
operations
with the greatest
γiiq)
The proof of this consists of the fact that the operations %'δ; = y£ are obtained as linear combinations of the operations kn (Ψ — 1) by virtue of the condition γ^ - 0 for i > 0, where re is large,
and the determinant of the transition from the kn(x\!k — 1) to the xl8t
is equal to 1. In fact, the process
ι
described above for constructing {χ $ϊ) is the process of reduction of the set of transformations
knCVk— 1) to the set y ; of "triangular type" on Ζ [χ] = k (Ρ). More exactly: letrebe sufficiently
large that y . ( ^ = 0 for / < i and y . ( ^ = y^ for i < f(n), where /(re) -> °» asre-> °° and y. n = Σλ^.^^Ψ^).
Under the condition Σλ^η?Α" = 0, one can write all these operations in the form Έ.μ^η^kn(Vk — 1)
and then apply to the set /ε"(Ψ*— 1) the process of reduction to "triangular form" described above for
constructing the operations {xl8i) up to high dimensions. We assert that the passage from !ΑΠ(Ψ — 1)!
to \γιιη\ is invertible. Indeed, any operation of the form Σλ^.Α:™Ψ has the form ΐιίγί>η + bl, where
•b[V = b[1*? = 0. Hence the operation i t has the form bl = μ2γ2,η + b2, where b2V •- b$ = b[2* = 0, etc.
866
S. P. NOVIKOV
Consequently, a =
^ j μίΥί + bf(n),
where b S'^
= 0, / < fn.
If re -» °o; then /(re) -» oo and the
coefficients μ£ stabilize, while an = 1iîxl8i + bf(n) if ο = (α η ) €βψ. Since the greatest common divisor of the homomorphisms a*1 , for all a GB^. such that a* = 0, is invariant and this invariant
can be calculated with respect to any basis of operations in Β
such that a*
= 0 , we have that for
l
the basis (x 8j) = (γέ) it coincides with the greatest common divisor for the basis (k"C¥k—l)) = bk,
where b^'* = kn (k' — 1). We note that the operations (knXVk) ate nonstable, but, by virtue of what has
been said, there exist operations o^.) n such that a^J^* = kn +' for / < /(re), where /(re) -> oo as η -» <Ό.
These operations are obtained by the transformation from (x 1 ^) to (λ"Ψ ) inverse to that described
above.
Remark 8.1. The same operations Ψ in k* ® Q are obtained as formal sums of the form Σμίχίδ£
= Ψ4, where μί € Q and A > ; € Ζ for large re and i < /(re).
Example 1. Let X = Ρ 6 Ό be the point spectrum. Then k* (P) has a single generator t as an ^ψmodule and is given by the relations δ;(ί) = 0, i > 0. The module k*(P), as a βψ. -module, has a single
generator f and is given by the relations (ρ"Ψρ)(ί) = pnt for all primes ρ (re large). (Or: all operations
a € βψ which have zeros of order one are such that at = 0.)
Example 2. Let X = MUn. Then k* (MUn) can be described by the ideal in the ring of symmetric
polynomials in the ring Λ [u1, . . . , un], dim ut = 2i, generated by u = ut . . . un. Let νi = xuit Ψ (νι )
A
= xHk(x)xVk(y). The elements of k*(MUn) have the form Σλ;> sxsds, where
= ((v. + 1) -1)' and Ψ(χγ)
ds = f(ul7 . . . , un) is an element of the symmetric ideal in Ζ [ul, . . . , un\ generated by u = ut . . . un,
and χ is the Bott operator. This uniquely determines k*(MUn) and k*{MU) as Ay. - and By. -modules.
We have the following.
Lemma 8.2. The ring B^ C.4*. coincides exactly with the subring of Αψ consisting of operations
of dimension < 0.
The only thing which must be proved is that Βψ contains all operations of dimension < 0. For a
pair al € Bk , a2 € B^. of operations which have zeros of order qlt q2 respectively, we introduce the
operations x'Qia1 = b1 and x'Qla2 = b2 and the composition b1 ob2 in Αψ . We shall show that xQl +92bl
o b2liesinB^
ifx^b.eB^.
L e t aln = ^
λ^knWh
and
a2n = Σ μ^^ΨΚ
We consider
)(
h
q2 q2X
k
x
k
12
iusing k x V
= H x' }.
m = η — q2. Then
h
We shall assume that re is very large, re -> oo, qi and q2 are fixed. We set
2ΑΛ
m)
η)
η)
where A^ = λ^ and μΙ
q
= k μ^ • Clearly, a s m »
867
we have a composition of operations in Βψ
which lies in B^. .
By virtue of the lemma, the rings βψ and <4ψ contain operations which coincide up to dimensions
f(n)
ε
n+
-> °c (as η -> °°) with the operations (Α"Ψ' ) in the sense that a-kpn,* = k
This remark allows us to use (up to any dimension) the ring Βψ
ρ
η
k
by (ρ"Ψ ), with ρ prime, and by χ e B
w
ρ
η
> for ; < /(re).
as if it were the ring generated
ρ
ρ
, where ίρ Ψ )χ = ρχ(ρ Ψ ) and y p = (ρ"Ψ ) are polynomial
k
generators. Thus, a (topological) basis here is x P(y2, y 3 , . . .), where Ρ is a polynomial.
We consider the βψ -module k*(P). We have
1
Lemma 8.3. The torsion part of the group Ext '£'(k* (P), k* (P)) is a cyclic group, whose order
l
is equal to ip"(p —l)! p , where η is large, ρ is prime, and \ \p means the greatest common divisor of
the sequence of integers.
Proof. We construct a Βψ -free resolution of the module k* (P). Let η be large. Then the module
k*(P)
is given by the relations (yp—pn)t = 0. We choose generators κ ρ = (yp—pn) and 1. Then the
κρ are polynomial elements,
Co = Βψ,
Ci = ΑΒψ,ρ,
Ρ
while du = t and dup = Kp(u), where u, up are free generators of the modules B^ = C o , β ^ _p C C l ;
respectively.
We consider the complex Honijt (C, k*(P)).
Let ht € Horn21 k (Co, k*(P)) b e ^ l e m e n t s such that hL{u) = x\t), and ht{p) C H o m ^ C , , k*(P)) be
s u c h t h a t hlP\up)B^x\t)
a n d 0 = hi(P\up^)
for p ' ^ p .
Obviously, we have
{ d * h u llp) = (hi, Kpll) = χρχί (t)
Hence, d*hi=
^\Ρη(ρτ—l)î
· Thus,
0 »»"(( ρρ*ΐΐ)) > ,
Since Hom*( , k*(P)) is a free abelian group, the element
of finite order equal to dt in the group Ext 1 ^ 2 1
i s a 1-cocycle for the operator d*.
d{hi)
.
—:
,
is the unique element
\P \Pl 1 ) / P
(k*(P), k*(P)), dt = ίρ^ρ'-Ι)!,,, η -^ «..
Note that the computation of E x t A (k*(P), k*(P)) presents no difficulties, since the module
ψ
k*(P) has a βψ-free resolution which coincides with the complex for the polynomial algebra Z[y 2 , . . . ,
y p , . . . ] , as long as the operator χ acts freely on k*(P), B^, .
We have
Theorem 8.1. The groups Ext l'*l(k*
η is large.
™
(P), k* (P)) are cyclic groups of order dt = \pn(pl— l ) i p , where
868
S. P. NOVIKOV
Proof. It is easy to see that the algebra Α ψ ®<2 is precisely the algebra of operations in k-
theory k* ® Q. Hence, by virtue of §2, we have:
* (P)®Q, V (P) ®Q) = E x t ^ (A (P), k(P)®Q) = 0
21
for s > 0. Hence the groups Extjj'/, are all torsion. We consider the resolution
Ψ
I
d π
h £ .k β t
\
ί ...->• 2iAw,i->-A<qr-+-k (Ρ) J = C,
i
where o?(u;) = Sj(u), and uiy u are free generators of C t and Co.
We consider the nonacyclic complex
where d(Vj) = xl8i(v),
and vi7 ν are free generators. We shall show that the complex C is such that
in the group
the torsion part is exactly the same as in the group
In fact, if λ;· 6 Hom^^ (C, k* (P)), where hj{v) = x J (i), then
Thus, if <i*Ay = Σ/ί^-ίΊ, where A ^ / ^ i ) = xhi(t),
where AJ- (νέ) = x'(t) and the numbers μγ
then
are the same. We note that the order of the group
is precisely the greatest common divisor of the numbers μγ
as i varies, and a generator is
l
\'^\i. Since the elements (x St) give a system of relations in k* (P) over the ring Βψ C Α
give the torsion part of Ext'n2/, since the complex C over Α-ψ is a segment of
ψ
a Βψ -resolution of the module k*(P). By virtue of the lemma, we get the required result. The theorem
the same integers μ/
is proved.
We now pass to the module k*{MU).
We have
Theorem 8.2. For any complex X € D there is a canonical isomorphism
Hom>v (*' (MU), k* (X)) = U* (X).
The proof of this assertion is e s s e n t i a l l y a straightforward consequence of the result of L 2 2 J
concerning the fact that the Riemann-Roch theorem on the integrality of the number ch ζΤ{Χ) ιΧ\ gives
869
a complete set of congruence relations on Chern numbers in Ωμ. More precisely: if LA j 6 il/j is an
indivisible element, then there exists ξ € K(X) such that ch ξ Τ (Χ) [Χ] = 1. By virtue of the properties of the Thom isomorphism in K-theory, this assertion is equivalent to the following: for any indivisible element α € n*(MU), there exists ξ 6 K"(MU) such that ( c h £ Ha) = 1, where Η: π* -» //*
is the Hurewicz homomorphism. Let β € Horn*. k (k*(MU), k* (P)); then the number (ch<f, /3) is also
ψ
an integer by virtue of Bott periodicity. Both groups Hom^j (k*(MU), k*(P)) and n*(MU) have no
torsion. iNote that Horn , k (k*(MU), k* (P)) C k* (P), for k* (MU) is cyclic on un.\ Hence π* {Mil)
IT)·
C Horn , . By virtue of what was said about the indivisibility of the numbers (ch ζ, Ha) a € ττ* (MU),
the group n* (MU) is indivisible in Horn , . Since the ranks of these groups coincide, the groups
coincide. Thus the assertion is proved for the point spectrum.
Let X € D, Χ^Χχ € 0, with A\ a skeleton of X, X2 = Χ/Χχ; we have exact sequences:
-£/· (Χι)-»-(),
Ο — ft' (Xj)-> ft* (X)-»- ft* (Χι)-»- 0.
We assume by induction that the theorem has been proved for A\ and X2 (we do induction on the rank
of the group //* (X, Z)). Then we have a commutative diagram of exact sequences:
. TJ* ι γ \
,_ TJ* ι γ\
—*• U (Λ-%) —>U \Λ.)
I))
vhi
0—» Horn*
ft
"IB·
. rτ* i v \
. Λ
—^(7 \A-l)—*• "
ν I))
—> Horn* Λ —» Horn* h —>· E x t 1 ' ! (k* (MU),
"ψ*
-^-VL·*
"ΪΙΛ
ft*(X2)).
However, by virtue of the commutativity of the diagram we have that the homomorphism
HomA^ (k* (MU), k" {X) )->-Hom* A ^ (ft*(MU), ft* (X 2 ))
is an epimorphism, since i is an epimorphism and γ is an isomorphism. Hence the homomorphism δ
is trivial, and hence by the 5-lemma the homomorphism ν is an isomorphism. The theorem is proved.
Remark 8.2. In what follows it will become clear that the groups Ext 1 '! (k (MU), k*(P))arenonΑ"ψ
„
trivial even for t = 1, and the question of their computation is extraordinarily important (see §9, 11).
By analogy with the rings A^. and βψ it is possible to construct analogous rings A^r and
Let kO be the theory defined by the spectrum kO such that Q,s'lkOn = BO χ Ζ (see §3). The cohomology ring of a point kO (P) = AQ is described as follows: generators 1 € Λ° h € Λ Ο ι ν ε AQ,
w Ε A'o; relations Ih = 0, h1 = 0, hv = 0, v2 = Aw.
We have the "complexification" operator
c: kO'^k'
such that c (h) = 0, c (v) = 2x2, c (w) = %4, where χ is the Bott periodicity operator.
nX
k
In the theory kO* it is possible by analogy with the theory k* to introduce operations (k \! ) and
their combinations α = (o n ), an = Σλ^" (JtnX9k), where α$ does not depend on re. The ring of such
operators is identical to the analogous ring for A;*-theory which lies in fi γ . The ring Β ψ is composed, in a fashion identical to that for the ring Βψ, from such operators a = (an) constructed from
Ψ and from the multiplication operators on Ao = kO*(P), keeping in mind the following commutativity
relations: Ψ*Α = khWk; Ψ*υ = k2vVk; Vkw = k"w^k.
We denote the resulting ring by S^° . Similarly,
870
S. P. NOVIKOV
it is possible to construct a ring Α ψ also, but we shall not consider this ring in what follows.
We consider the category Β C D C S.
1) The spectral sequence (En dr) j kO* is trivial in B; in Β there is a subcategory β ' s u c h that:
2) the operation of the ring fi^ is well-defined in B'. As is easy to see, the spheres S^their
spectra in S) lie in B' by definition, since kO* (Sn) = W*(P).
If f:Sn+k-*Sn
UfS"
is a mapping, then a necessary and sufficient condition for the complex Dn
n
to belong to B' is that f* = 0, f :W*(S )
-» W* (S
In the category Β D B' the operation of the ring BL
extension Βψ
+
k+1
n+ k
).
is well-defined, the latter being α priori an
-» β ψ , since in view of the presence of torsion in AQ — kO*{P) the operation is not de-
fined by its own representation on kO*(P), in contrast to &*-theory in the category D.
There is defined a homomorphism (epimorphism):
yk0 (kO* (P), kO* (P)) -y ExtÎo
ψ
"ψ
and a Hopf invariant
qx: Ker? 0 — E x t K 0 (ΛΟ* (Ρ),
It is easy to see that the complexification c:kO* -> A* is an algebraic functor (see Definition 9.1)
from the category of βψ -modules to the category of Β ψ
-modules.
It is also easy to show that
1,4ft
E x t i # ( k O * ( P ) , W(P))=Zdh,
dh=
-
{pn(p^
i))p
and
Extsô (Λο,Αο) = Z2 for s = 8A + 1, 8k + 2.
ψ
We have a natural ring homomorphism τ : Β ψ
-> S,j, generated by the homomorphism c:kO*(P) -*k*(P),
and consequently a homomorphism
1,4A
1,4 4
c : Ext B ^° (Λ ο , Λο)->- E x t ^ (Λ, Λ),
A = k'(P),
Ao=W(P)i
whose image has, as is easy to see, index 1 for k = 2/ and index 2 for k = 2l + 1, in consequence of
the fact that the image of the homomorphism c: kO1 -* kl has index 1 for t = 81 and 2 for ί = 8/ + 4.
Later, in v9, this homomorphism will be considered from another point of view.
by Λ
There is defined an element h € Ext 0 ' 1 un (AQ, AQ) such that 2h = 0, h3 = 0, while multiplication
nk
ν
0,8h+l
Ext B f e o
is a monomorphism for s = 8k, 8k + 1.
h
-
0,8ft+2
l,s
l,s+l
^
0
871
The images of the homomorphisms
0,·
and
are easy to study: namely, q0 is an epimorphism (see L^J), and the image Im <jr, is realized by the
n
image of qv ο J, where J -n^SO) -> 77*(S ), and i s nontrivial in dimensions (1, 4k), (l,8k+ 1), (1, 8k + 2).
§9. Relations between different cohomology theories. Generalized Hopf invariant.
i/-cobordism, Α-theory, Z p *cohomology
Let Χ β S be a cohomology theory. Suppose given a subcategory Β C S. We define the notion of
the "Steenrod ring" Α β of the theory X
in the subcategory B: the ring Α β is the set of transformations
θ κ : Χ (Κ) -> Χ (Κ) which commute with the morphisms of the category Β (according to Serre). The
tln
S
Αβ contains the factor-ring A /] (B), where J (B) consists of all operations which vanish on the
category B.
We now define "the generalized Hopf invariant:" let
g: K^KZ
be a morphism in Β such that the object CKi [j K2 ( = 0 + j K, in the notation of § 1 , i.e., the sum
with respect to the inclusions K1 — and Kl ^> K2) also lies in B.
We have an exact sequence
·
υ κ2)
C
If the homomorphism g = q0 (g): X (K2) -· {EK^ is trivial, then we have
Ο - ν Γ ( ί , ) ^ Χ * (CKi U K2) -*- X ' (K2) -*• 0»
g
where Χ (Κι), Χ (CKl (J K2) are modules, and our short exact sequence determines a unique element
g
?i(g)eExt A *
(Χ*(Ki),X*(K2)).
We thus obtain a mapping
(Β)
q± : Ker ςτ0 ->- E x t i f (Χ* ί^), X* (K2)),
where qa:Hom* (Klt K2) -. Horn* Α" (Λ7* (K 2 ), A:*(K1)), K,, K2 € β and g 6 Kerq(0B\
provided CK,
U K2 € Β. This map is "generalized Hopf invariant."
S
-1-
General problem: which elements of E x t ' ' x (^ (^2). % (Xj) are realized geometrically as
images ql (Ker qoB))?
If Α β ε .4 β is an arbitrary subring, then there is defined the usual homomorphism:
872
S. P. NOVIKOV
i: E X U * (X* (K2), X* {Kx) )-^ E x t " f (X* (K2)), X* (K,))
and we set q1 = iqi, where qy is the "Hopf invariant" of the subring -4g C /Ιβ.
Examples.
If Β consists of a single object K, then .4g = YLndX* {K) and there is no Hopf invariant.
2. If Β consists of objects Klt K2, L = CK1 \J K2 and morphisms g : Kt -> X2, /3 : L^ EKY,
α : /ϊ2-.
L, where g : Z (£ 2 ) -> Α" (/Ϊ,) is the trivial homomorphism, then the ring /4g consists of all
endomorphisms of X (L) which preserve the image β* X* (K^ C X (L). In this case, the Hopf invariant qiB)(g)
€ Ext 1 X* (!*(&,), X*(K2)) is defined, and is equal to zero if and only if X* (L)
Β
= X (Xj) + X (K2) (as groups). Of course, examples 1 and 2 are uninteresting. We go on now to the
examples which interest us.
3. Let Β = D (complexes with no torsion) and X* = H* ( , Z p ) . In this case A% = Α/(βΑ
+ Αβ),
where β is the Bokstein hombmorphism and A is the Steenrod algebra (over Z p ) .
There is a canonical isomorphism
ExtA {Ζρ,Ζρ) — ExtΑ/$Α+Α$(Ζρ,Ζρ)
for t > 1, where Z p = Η (Ρ, Ζρ), Ρ is a point, and the Hopf invariant qi (D) coincides with the Hopf
invariant ql for Xt = K2 = P.
4. Let Β = D and .Y* = k*. In this case A\ D /4^, and the latter ring contains the ring A
/](B)
but apparently does not coincide with it. The Hopf invariant in this theory will be discussed later; the
Ext1-/* (k* (P), k* (P)) were computed in § 8 , In §8 we considered the subring Βψ C Ak
and
ExtA*, (k' (P), k* (P)) = tor ExtBV (k' (P), k' (P)).
5. For the theory X
= U we shall also consider the category Β = D and the Hopf invariant for
the whole ring A .
The groups Ext 1 · {j will be computed later (for K2 = MSU; see §6).
A.
6. In §2 it was indicated that for complexes Κ = E2L the homomorphism / : K° (X) — I (X) can be
considered as a homorphism / : K° (X) -> Ρ (X), where Ρ is the point spectrum or cohomotopy theory.
A lower bound for the groups / (X) can be computed in any cohomology theory Υ , if we consider the
composition
q[Y)-J: Κ°(Χ)-+Ρ'(Χ)-+Εχν·*(Υ"(Ρ), Y'(X)),
AY
where P* (X) = Horn* (X, P), defined on elements such that q(0Y)
• } = 0.
If Κ = EL, then in this case the computation can also be carried out by means of Ext
Υ
γ (Υ
(Ρ),
(Χ)), but here the multiplicative structure in Ext**y enters by virtue of Lemma 2.1 of §2.
We now consider two cohomology theories Χ , Υ
€. S, a subcategory Β C S and a transformation
α : X* ^ Y* of the cohomology functors in the subcategory B. Let subrings ^g C /lg, Ag C Ag be
chosen.
Definition 9-1· We call the transformation α : Χ
^ Υ
algebraic with respect to the subrings
Λβ> ^ S , if it induces a functor α from the category of ^g-modules to the category of ,4g-modules.
When AB = Αβ, and Ag = Αβ, we call the transformation α algebraic.
873
Examples.
1. Let Χ , Υ
be arbitrary cohomology theories. An arbitrary element α € Υ (X) determines a
transformation of theories
a: X* -> Y'.
2. If the theory X* is such that X1 (P) = 0 for i > 0 and Χ" (Ρ) = π, then there arises an augmentation functor
ν: Χ*^Η*(Υ,π)
and hence for any group G a functor
, π®
yG: X'-+H'(
For example, for G = Zp we have vp:X
G).
— Η ( , π ® Z p ) . In the c a s e s of interest to u s , π = Ζ and
rr<8)Z p = Z p .
3. The Riemann-Roch functor. Let X
η)
Grothendieck element λ[
€ K°(MUn).
= U and 7
= A: ; we consider the Atiyah-Hirzebruch-
It defines a map
λ_ι:
U*-+K'
and λ : i/* -. A*, where λ = (λ ( / ι ) ), λ ( π ) Ε k2n (MUn), is the element (uniquely defined) such that %"λ ( π )
= 1 6 K°(MUn), where* is the Bott operator.
For the theory X
= U , the augmentation functors v, vp
and the Riemann-Roch functor λ preserve
the ring structure of the theory.
Later it will be shown that these functors are algebraic in the category D.
Now let a: X
-> Υ
be an algebraic transformation of theories in the category Β C S with respect
to the subrings Ag, Α β. What is the connection between the "Hopf invariants" q\
in the theories X
and Y*?
Since a:X
-> Υ
leads to a functor in the category of modules, the trivial morphism βχ : X
(K2)
-> X (Kt) corresponds to the trivial morphism gy : Υ (K2) -· Υ (/£,) for /^, Κ2, g €, B. Hence we have
the inclusion Ker <7οχ C Ker <?oy >
an
d the domain of definition of the Hopf invariant ql^
is contained
in the domain of definition of (f/y .
Now let α be a right exact functor in the category of modules. We consider a resolution Cx of the
module Μ = X (K2) and the following (commutative) diagram:
CY Z-
CYâCx
where Cy is an acyclic ^g-free resolution of the module α Μ = Υ (Κ2), Gy is a free complex such that
H0{CY)=~aM.
Let Ν = Χ* (Κ,), α/V = Υ* (Κ,).
By definition we have: Η* (Hom^ y (aCx,
Β
a,V)) = R*GN(M), where R* =
^ Rq
ar
»d GN =
4
Hom^V ( , a.N)°a. is the composite functor, RqG is the g-th right derived functor. There is defined a
Β
874
S. P. NOVIKOV
natural homomorphism
rq: Ext^x {M, N) -> R"GN (M),
r = 2 rq,
A
B
q
and homomorphisms
Pi': RqGN (M) -> Hq* (Hornby (CY, aiV))
A
B
(
y(aAf,
aTV),
A
B
where Ker /82* = 0.
We have the composite map
a=
(tir^W.
E1(a)-*Ext1-7(Y*(Kt),
7·(ΛΓχ)),
A
B
where
£x (a) c E x t U (X* (tf2), X* (^)),
A
B
Ex (a) = Γ^οβΓ'βϊ ( E x t U (Γ* (ΛΓΟ. i " (î))·
A
B
In the following cases the group El(a) coincides with the whole group Ext:
a) α is an exact functor; here H'(aCx)
= 0, i > 0, and one can assume that Cy = Cy, /82 = 1.
b) If in addition a is such that Ext^y ( , c7V) °e = 0 for i > 0, then aC^- = Cy and an isomorphism
is generated:
ExtlY
(άΜ, aTV) = Λ*(5Ν (Μ).
In case (a) (a is an exact functor) there arises a spectral sequence (Er, dT), where Εζ'
RpGq,N(M),
— 2
P,
which converges to Ext** (aM, a/V), and GQi
N
q
(M) = Ext?. * ( , e\')°c. From this
Q
spectral sequence it follows immediately that the homomorphism
is a monomorphism.
The basic examples which we shall consider are the subcategory D of torsion-free complexes, the
theories U , k , Η ( , Z p ) , the Riemann-Roch functor λ : U — k
and the augmentations vp : U
-> H*( , Z p ) . We have
L e m m a 9 . 1 . a ) T h e functors
category
λ : U -> k
and v p : U
- > / / ( , Z p ) are algebraic
in the sub-
D;
b ) The functors λ and up are exact in this category.
c) The functor λ is such that RqGN (M) = Ext ?
υ
{Μ, Ν), where Μ = U* (K2), Ν = U*(KX), Μ, Ν € D, GN
î ( , λ/ν)°λ, λΝ = k*(Kj.
d) The functor
vp
is such
that RqG^(M)
- Ext^ ,»,
^»(i/pM,
i'pN).
875
Proof. The category of A -modules corresponding to the category D is the category of Λ-free
modules, where Λ = U* (P) » Sly. On the cohomology of a point Λ the functor λ is such that Λ - Ζ [xl
and \(y) = T(y)x\ where y Ε Ω|/ = V'2i (Ρ) and Τ is the Todd genus.
From the group point of view we have \M = Μ & Λ Ζ [χ], where Μ is Λ-free· There follows the
exactness of the functor λ and Rq\ = 0, q > 0. For vp we have vpll (P) = Z p , and in the category D,
vpM = Μ ®ΛΖΡ; since in the category D all groups U (K) and // (K) are free abelian, the functor vp
is exact in this category. This proves part (d). Part (c) follows immediately from the theorem in §7.
Part (b) follows from the well-known fact that H* (MU, Ζp) is a free (Λ/βΑ + /J/S)-module. We shall
now prove the fundamental part (a).
Consider first the functor λ. We recall that in §5 we constructed operations Ψ^ £. A <8> Q. Let
Ψ" (λχ) = λψΐ (χ),
ΐ ε IT (Κ)t
where Κ €. D is a complex with no torsion. Since λ is an epimorphism and A(y) = Τ(y)xl, where * is
the Bott operator, the desired formula follows easily from the construction of the Adams operations
Ψ*' in K-theory and of the operations Ψ/y in §5. The operations (knrVk) have the form kn\^l, and are
"integral" for large n. Thus, the action of the operators (&"Ψ ) and multiplication by % in A: -theory
are calculated by A and λ. This proves part (a) of the lemma for the functor λ.
Now let α = vp : V* -> H* ( , Z p ) . In §5 we constructed a projector Φ 6 Αυ® ζ QP of the theory
U onto a smaller theory having the cohomology of a point Λ ρ = Qp lx1, . . . , x^, . . . \, dim%;=— 2(p' — 1).
We set
Pk (vpx) = ν Ρ Φ5 ω Φ (χ),
where ω = (ρ - 1, . . . , ρ — 1) (h times) and the Ρ are the Steenrod powers. The correctness of this
formula follows from the fact that all homomorphisms (Φ5ωΦ) (γ) = 0 mod ρ if dim ω = dim y, i.e.,
(Φ5ωΦ)* (y) € Ω[/ = Ζ. The lemma is proved.
Corollary 9.1. For any Klt K2 € D the homomorphism
α — λ: ExtAV(U* (K2), U* {K,) )-*• Ext% (k* (K2), k* {K,))
is a monomorphism.
Proof. As was established in Theorem 8.2, the homomorphism r ; is an isomorphism; the homomorphism β* is a monomorphism, as was shown above, while β* = 1, since Rq\ = 0, q > 0. Hence,
β* rl = λ is a monomorphism.
Corollary 9.2. For any complex Κ = E2L the. lower bound of the J-functor
coincides with the bound
<
q $-J(K°(X))(EEExtTv(U*(P),U'(X)).
Corollary 9-2 follows from Corollary 9.1.
Corollary 9.3. The groups Ext^'y 1 (U* (P), U* (P)) are cyclic groups — subgroups of cyclic groups
of order equal to the greatest common divisor of the integers \kn (kl — l)\k for all k, for large n.
S. P. NOVIKOV
876
1
1
Proof. Since the groups Ext .'! (A (P), k (P)) by virtue of the theorem are cyclic of the asserted
Ψ
orders, Corollary 9.3 follows from Corollary 9·1·
We shall indicate a simple fact about the connection between the Hopf invariants in different
cohomology theories Χ , Υ in the presence of an algebraic transformation a: X -. Υ with respect
to the rings Ag, Α β in the subcategory Β C S.
Lemma 9.2. We have the equality
~ _(B)
_(B)
qiY = a-qix
on Ker ?o^\ the group ^^'(Ker qi^)
being contained in Ei(a),
the domain of definition of the homo-
1
morphism a. = (β*' · β* · rr).
The proof of this lemma follows immediately from the fact that by construction of the generalized
Hopf invariants qlx
and q^Y
a€ Ker ^ ' C K e r
q0(p.
we can compute both quantities qfy
(a) and qfy
(a) for any
As is easy to see, the equality
is true. This equation is equivalent to everything asserted by the lemma. The lemma is proved.
Corollary 9.4. a) If the element γ € ExtL'χ does not belong to Ei(a) for any algebraic a.: X
-. Υ
A
B
then the element γ is not realized as the Hopf invariant of any element of Horn {Klr K2).
b) // γ € Ext\iy does not belong to the image of the homomorphism a(Ext!_y) a-nd Ker qiv'' =
Λ β
B
Ker gay » then the element γ is not realized as the Hopf invariant of any element of Horn* (K1, K2).
§10. Computation of Ext1 U(U*(P),
U* (P)). Computation of Hopf invariants
in certain theories
In the preceding section the monomorphicity of the mapping
Εκύυ (U* (P), U* (P))_- Exti» (k* (P), k* (P))
was established.
We shall now bound the order of the groups Ext1'?,1 from below. We consider the resolution
where Co = AU (generated by ίί) and C\= 2_,Αω with generators ίΐω
άυ,ω= 5 ω (ω), dim ω > 0. We con-
ω
sider the differential
ω>0
where
d* (x) =
2 Υ*ω (x),
x
^ ®u = Hom 2 l« (Co, Λ)
877
and
(C\)
where σ%{χ) ίαω']
= 0 if ω Φ ω ', and σ&Ο) luM]
> Ω
= σ*ω{χ) £ Λ. These facts follow from §5.
Now let i be odd. We consider the element xj, where x1 = ICP1} € Ω|/. Since σ*{χχ) = ±2, all
θω( {) = 0 mod 2, ω > 0, from the properties of the homomorphisms σ*ω described in §5. Hence the
cokernel Coker d* always contains an element of order 2. Since the homomorphism Ext^'y"1"2
-» Ext 1 '^
is monomorphic and \kn{ki — \)\]c = 2, i'= 1 (mod 2), we have
χ
Extl'"
+2
(C7* (P), J7* (P)) = Extl'^
+2
(ft* (P), ft* (P)) = Z,
for 4i + 2 = 2 t , i = l(mod 2).
Thus, we have proved
Theorem 10.1. The groups Ext'' 2 j(A, Λ) are isomorphic to Z2 for i = ll + 1.
We now study the case of even i = 2l. Let y ; £ Ω}} be an indivisible element such that some
multiple Ay i; λ φ 0, represents an almost-parallelizable manifold M 21 , whose tangent bundle τ is a
multiple of the basic element κ; of the group K°(S21), τ = μ; κ;, μ; integral, where κ; = / κ ; , /":Λ/
-· S 2 ' a projection of degree ± 1 . From the requirement of the integrality of the Todd genus and the
fact that (ch κ1, Μ21) = 1, it follows easily that all οωγι for all ω are divisible by the denominator of
the number ( β ^ / α ; · 2l), where a 2 s + i = 1 a n d a2s = 2, S; is the Bernoulli number entering into the Todd
genus, i = 2l (see L14J). Hence Coker d contains a group of order of equal of the denominator of
the number (Bi/af 21). Since this number is only half of the number \kn (kl — l)\ k (α ι = 1), the image
AExt 1>2i C Ext 1 · 2 ,' coincides with Ext 1 ' 2 ' for I = 2s and has index 2 in Ext1'2,1 for / = 2s + 1.
A<J
A^
A^
A^
From this, for the case o; = 2, I = 2s, follows the
Theorem 10.2. The groups Ext 1 ' 8 ^ (Λ, Λ) are isomorphic to the groups Ext 1 ' 8 ^ (k* (P), k* (P)).
In the case a[ = 1 there arises an uncertainty: do the groups Ext1^8
Ext*'^ + 4 or do they have index 2 in them?
+4
coincide with the groups
Hence, we have the weaker
Theorem 10.3· The groups Ext1^8
+4
( Λ , Λ) are cyclic groups whose order is equal to either the
denominator of the number S 2 /t+i/(4^ + 2) or the denominator of the number B2ii + 1/(8k + 4).
Remark 10.1. In what follows it will be established that this order is in fact equal to the denominator of B2jc + 1/(8k + 4) for A: > 1 (however, for k = 0 it is easy to see that Ext1'4,, (Λ, Λ) = Z 1 2 ). The
basis element u^. of the group Ext1<8y + 4 is such that
i,8fe
d3(uh) = h3ExW
1,2
(Λ,Λ),
fee
Ext A " (Λ,Λ) = Z2.
We now study the question of the relations among different cohomology theories and the question
of the existence of elements in the homotopy groups of spheres with given Hopf invariant
for the cases X = U , k , kO , //* ( , Z p ) , with the help of the functors α = λ, α = c, a = vp relating
these theories.
878
S. P. NOVIKOV
1. The first question which we consider here is the complexification
with respect to the rings Βψ and βψ. The structure of the groups
ExtB*,° (ft0*(P),ft0'(P)),
where s = 0,1 is known to us, namely:
a) Erti*o (W* (P), ftO" (P) ) = Z2,
t = 8ft + 1, 8ft + 2,
Extfifco (kO*(P),kO*(P))=0,
i=#8ft + l,
b) E r t i $ (ftO* (P), ftO" (P)) = Z{h«(ft*-i»fe + • • ·,
k ^ 0,
8ft + 2;
η -» oo,
Extiô +< (ftO* (P), ftO* (P)) = Z 2 + . . .
for f = 1,2;
c) the homomorphism q0 :π$(Ρ) -> Horn ^ 0 (ΛΟ (Ρ), kO (Ρ)) is an epimorphism (result of Brownψ
Peterson-Anderson L J);
d) the homomorphism <jrt · / : kO1 (P) -» E x t l f 5 0 (AO (Ρ), &0 (Ρ)) is an epimorphism. This last
fact follows from the work of Adams L J for the groups Ext 1 ' 4 ^ (Ag , Λο); since
ψ
l,8ft+i
Exts*°
l
l,8ft
= h Extfi*°
the required fact follows for the groups
(t = 1,2),
o,l
Α ε ExtBfco,
Ext1'*Aft(Ao,Ao).
We now consider the complexification c, defining homomorphisms ^c, "c ':
Exthko(Ao, Λο) — Ext^%(A, Λ) = Ext 1 ;** ,
^ i 0 (Λ ο , Λ ο ) — [
c
Since in the groups k (P) the image of the homomorphism c has index 2 for t = 8k + 4, index 1 for
t = 8k and is equal to zero for t^ 8k, 8k + 4,we can draw from this the conclusion that the image
group Im'c C Ext 1 '^ (Λ, Λ) has index 2 for I = 8k + 4, index 1 for I = 8k and is equal to zero for ι Φ 8k,
Α
ψ,
8k + 4, since I m ^ = Im'c'.
Consider the groups πη + *k-i (S n )and the Hopf invariants in kO - and k -theories. These invariants
are always defined since Ext 0 ' 4
= 0. We have thus the
Conclusion. The image of the Hopf invariant
1,4ft
quh:
t
nn+ik-i(S»)ÊxtA^(k (P),
h a s i n d e x 2 for k = 2l + 1 a n d i n d e x 1 for k = 2l. M o r e o v e r , t h e i m a g e q1>k(1Tn
the image quk • Ιπ^-ι (SO).
+ 4k-i(Sn))
c o i n c i d e s with
879
2. We now consider the Riemann-Roch functor λ : U' -> k and the corresponding homoraorphism
λ : Ext x 'J. (Λ, Λ) -· E x t 1 ' ! • Since λ is a monomorphism, we get from item 1 on complexification the
following conclusion:
The Hopf invariant ql : nn +ik-i (Sn) -> E x t 1 ' ^ is always defined, and its image Im qly coincides
with q^f, (J π2ι(.ι (SO)); it coincides with Ext 1 ' 4 ,, (Λ, Λ) for k = 1, k ~ il and has index 2 in the group
E x t 1 · 8 , ^ 4 for / > 1.
Later we shall study Ext 1 > e * + 2 and E x t 1 > 8 * + 6 .
u
u
A
A
3. We now consider the functor vp : U* - H* ( , Zp) and the corresponding Hopf invariant
Jt n+i _i (Sn), η -> oo, i > 1
Γ
^β(Ζ
ρ
, Zp) = E x t ^ ( Z p , Z p ) .
Since Ext 1 ' 1
( 2 D , Z D ) = Ext/i(ZD Z D ) for i > 1, this becomes the usual Hopf invariant. Since
Α/ βΑ + Αβ μ
^
HP
r
the homomorphism
qw:
/ j t 8 f t - i ( ^ O ) - ^ E x t ^ (Λ Λ Λ)
is an epimorphism, the question of the existence of elements with ordinary Hopf invariant equal to 1
reduces to the calculation of this invariant on the group /^ 4 i.,(S0). For example, let p = 2, and let
hi € Ext^'2 (Z2, Z2) be basis elements. Since hi? h2, hz are cycles for all Adams differentials and
represent elements in the groups Jn*(SO), it follows that, in view of the fact that Im / is closed under
composition, A; • h2 £ Ext^' * (Z2, Z2) must represent an element of q2jn* (SO) if hL represents an element of q1Jn*(SO). Moreover, since qJn^k-2 (SO) = 0, we have h2hi = 0 if A; € qlJn*(SO), since
h2 € qJn*(SO).
However, hi · h1 /= 0 for i > 4. We have thus the
Conclusion. For i > 4 the elements hi € Ext^'2 (Z2, Z2) do not belong to the image of the homomorphism
v2: E x t 1 ; *
( A , A ) ^
2 i
The case ρ > 2 is considered analogously.
In fact, we have the purely algebraic
Theorem 10.4. The image of the homomorphism
v p : bjxtjj
-ÊxtA/Αβ+βΑ (Ζρ,Ζρ)
is nontrivial only for i = 0, 1, 2 (p = 2) and for i = 0(p > 2).
4. We now consider the homomorphism
δ:
ExttJj(U'(P),Ut(P))Êxt'A*u(U'(MSU),U'(P)).
We assume that Κ 6 E x t 0 ^ (U* (MSU), Λ), γ Ε Ext0^ ({/* (MSU), Λ), and h €. Ext 1 ·^ ({/* (MSi/), Λ) are
elements such that άζ(Κ) = A3, and y 6 Ω^ is represented by an almost-parallelizable manifold. We
have
880
S. P. NOVIKOV
Lemma 10.1. All elements
of the form hn + l · Ke · ym, η > 0, m > 0, e = 0, 1, belong to Im δ.
Proof. Since h € Ιπιδ, it suffices to show that Ke · ym · h belongs to Im δ. For this it suffices to
establish that all homomorphisms σ*ω(χί· Κ€• ym) ate divisible by 2. It is easy to verify that σ%(χι),
σω(Κ) and ow(ym)
are divisible by 2. The general result follows from the Leibnitz formula
As was shown in §6, in the Adams spectral sequence for U (MSU) we have:
m
a) d3 (hKy )
4 m
^ 0,
=hy
Moreover, Brown-Peterson-Anderson showed in L J that elements of the form hym £ ^su
belong to
the image of the homomorphism 77* (5") -> π* (MSUn) by a direct construction of the elements.
We have thus the
Theorem 10.5· a) The groups Ext1'^.
(Λ, Λ) = Z 2 are cycles for all Adams differentials and
belong to the image of the Hopf invariant
qiu- nn+sk+i(Sn)-Êxtû
b) The groups Ext1'^,
+6
(Λ,Λ),
η->-oo.
(A, Λ) = Z2 are not cycles for the differential d3.
Remark 10.2. Since Ext1>4J + 2 = Ext1>4/ + 2 , the analogous facts hold also for A>theory, although
basis elements here are not related to the /-functor, in contrast to Ext 1 '^ (here, the elements go into
fiym under the homomorphism Ω6 -> Ως^,).
We summarize the results of this section:
1) The groups Εχ^'ί^Λ, Λ) were considered and also the associated homomorphisms
Extl:h*o{kO'(P),kO'(P))
ExtÛ (Α, Λ)-
Β γ
t
1; U
1
Ά
1i,kO
yk(
(Zp, Zp)
.(
u
where qiH
is the classical Hopf invariant, / is the Whitehead homomorphism, λ is the "Riemann-Roch"
functor, c is complexification, and vp is the augmentation of U -theory into Zp-cohomology theory.
2) The homomorphism Ext 1
υ
(Λ, Λ) - Ext1
y
(U* (MSU), Λ) was studied.
3) It was established which elements of all these groups Ext 1 are realized as images of the Hopf
invariant qx. In particular, for the groups E x t 1 ' " (Λ, Λ) this image
is trivial for I = 4k + 3;
is an epimorphism for t = Ak — 1, 4k; for I = 4k + 2 (k > 1) and t = 4k + 3 (k >_ 0) the Adams differential
d3:
ΕχΙ^(Λ,Λ)^Εχ^ + 2 (Λ, Λ)
l 2t
is nontrivial; it can be shown that d,(E 2' )
(see
3
881
for t = 4k + 2 (k > 1) and t = 4k + 3 (k > 0)
= h E\·"'*
§11).
4) The nonexistence of elements with classical Hopf invariant 1 is a consequence of the fact
1 2
that "ΰ2 (Ext ' J) = 0 for i > 4. Analogously for ρ > 2 (see
§12).
5. For l £ 8k• + 4, the fact of the following group isomorphism was established:
f
*"
ι (
for I = 4 this fact is false. For t = 8λ + 4, & > 1, it is true and will be proved later (see §11).
§ 1 1 . Cobordism theory in the category
1
Earlier, in § 5 , it was proved that in the algebra A
S®?Qp
<S> zQp there exists a projector Φ €. A
such that Φ(χ, γ) = Φ(%)Φ(ν) and ΙπιΦ* C Λ is the ring of polynomials in generators y,, . . . , y
<£)zQp
_ _ _;
dim γι = 2p'— 2, where the y ; are polynomial generators of the ring Λ = Ω(/<8> zQp such that the numbers σ* (y) € <?p are divisible by ρ and au {jCi = P> k = p 1 — 1. Moreover, a complete system of orthogonal projectors Φ ( 0 was constructed, Σ Φ ( ι ) = 1, φ ( ί ) · Φ < ' > = 0, ι'Φ j , where the Φ ( ί ) (U* <g> zQp)
isomorphic theories up to shift of dimensions. Hence, in the category S ®zQp
equal to the sum MU Λ: ^ j Eld^Mv
are
the spectrum ML·' is
, where ω is not p-adic ji.e., ω = (il7 . . . , i^), all iq 4 pr - 1
k
for any r, and d (ω) = Σ ί' |. If l p is the Steenrod ring of the spectrum \Ip, where /l p = Φ · -1 · Φ,
q ~1
then we have:
1) \ U ®zQp
=
^'^ ( Λ ρ ) i s the appropriately graded ring of infinite matrices of the form ( α ω . ω .),
α
ω -ω- ε /Ι ρ , ωι not p-adic and dim (αω , ω .) = 2(/(ω;) — 2ί/(ω;) + dim αΜ.ω.
Ii.e., the right-hand side
is a constant for the whole matrix and defines the degree of the matrix!.
2) Exty
(U*(K), U*(L))®zQp
= Extyy
defined by the spectrum Mp.
(Ul(K), 6'*(L)), where U* = Φ(ί/*®ζ<?Ρ) is the theory
η
3) The Adams spectral sequences (Er®Qp,
dT ®QP)
in {/-theory and (Er, dr) in i/ p -theory
coincide. T h e s e facts follow from § § 1 - 3 .
We note that the polynomial generators of the ring Λ ρ = Up (Ρ) = Φ U (P) can be chosen to be
polynomials with rational coefficients in the elements xi = [CPP
"i
€ Ω^, where the polynomial
generator can be identified with LCP''" 1 ] - χΛ = y, in the first nontrivial dimension, equal to ρ — 1.
We consider the ring A P j ; C Λ, generated by the first i polynomial generators y 1 ; . . . , y; ε Λ ρ .
This ring Λ ρ > ΐ does not depend on the choice of generators.
The following fact is clear: the subring Λ ρ > ; C Λ ρ is invariant with respect to the action of all
operations Φ · 5 2 · Φ on the ring Λ ρ . The proof follows from the fact that the subring Λ
C Λ = Ω(/,
generated by all generators of dimension < 2/, is invariant with respect to S a ! and with respect to Φ,
We consider the projection operator Φ; ε Αρ ®ζ) Q such that Φ* : Λ ρ — Λ ρ ί , Φ ; | Λ ρ > Ι = 1 and
Φt(y;•) = 0 for / > i. The ring Φ;.4ρΦ; will be denoted by l p > ; . It is generated by the operators of
multiplication by elements of Λ ρ > , C A ; ; and by operators of the form Φ ί · Φ · 5 ω · φ · φ ί ; where it is sufficient to take only partitions ω = (kl, . . . , ks), kj = pi} - 1, while ςτ-; < i.
882
S. P. NOVIKOV
We have the following general fact.
The ring Ap
is generated by operators of the form Φ· SM· Φ for ω = (&,, . . . , ks), kj = pq> - 1.
This fact follows easily from properties of the projector Φ and the structure of the spectrum Mp.
However, if ω = (pQi — 1, . . . , p9s — 1) and at least one q, > i, then clearly σ ω ( Λ ρ > ; ) = 0. Hence
it suffices to consider only Φ, · Φ · 5 ω · Φ · Φ; for ω = ( ρ Ρ ι - 1, . . . , pqs
in the ring Φ ^ ^ Φ έ
- 1), where
all <7-; < i.
Additive bases for the rings Ap
a
and
Ap>i:
Λ
) Ap
= (Λ ρ · δ ω ) , w h e r e ω is p-adic and / \ denotes completion (by formal series).
t>) ΛΡι1
= (ΑΡι1.8ω)Λ,
ω
= ( ρ ' ι - 1 , · · · , pis-D,jk
We consider the operations e i ; ^ = S, i.
i.e., eiik
APil,
= Φ ^ α , Φ Φ ; € AptL.
1) Δ(βί,&) =
<i.
(^ times), regarded as elements of the ring
ι
Clearly, we have:
(the projectors Φ; and Φ preserve the diagonal);
2j eis®6i:i
h=l+s
2 ) e ? > t ( A p > i . 1 ) = 0;
3) e j i (γι) = ρ, where yi G A p > j C Λ ρ is the polynomial generator of dimension pl — 1.
We denote by fe; C APti,
the subring generated by the elements (e^ ^), A: > 1.
We denote by Dj the subring of APii
erator yit
spanned by &i and the operator of multiplication by the gen-
i.e., Dt = Cp [ y j &t.
We have the following
Lemma 1 1 . 1 . a) The subring fej commutes
operators
with all operators
Φ^Φδα,ΦΦ; /or οΖί ω = ( ρ ; Ί - 1, . . . , p / s - 1), zÂere j
b) /« ί/te rireg &j we have the
k
of multiplication
A.piimi
C Λ ρ are</ a//
<i - I.
relations
f
ei,h • 6i,s =
+
I
\
ei,h-yqt=
2
s
/
) · βj , ; l + S T
el.
s-\-m=i
where
e*m{yi)=
(
Q
c) The ring APii.i
then factoring
of ΒPj;,
) p™yV~™.
is obtained from the ring APti
the remaining subring BPli
by discarding
the polynomial generator y^ and
C .4 p > ! : by the ideal spanned by the central subalgebra fe;
where
Bp, i = {ΛΡ, f_j · (Φ,Φ&,ΦΦ,) } Λ
for all p-adic ω·
The proof of all parts of Lemma 11.1 follows easily from what has preceded.
Thus, the ring APii
is obtained from the ring l p ,;-i in the following way (in two steps):
Step 1. Without altering the "ring of scalars" Apii.lt
AP.i-·
we make a central extension of fe; + 1 by
883
with Q>i acting trivially on Λ ρ ^ ^ .
Step 2. We adjoin to the ring of scalars APtiml
setting eiik(yj)
The
a polynomial generator ji of dimension p' — 1,
= (pp/cyq-Zc with all the consequences derived from this.
ring Qp [yt\ · Bp,i coincides with Ίρ,ί, while the commutation rules for y; and Φί5ωΦί are
derived from part (b) of Lemma 11.1.
In particular, the action of the operators Φ;5 ω Φ; for ω = (ρ'1 — 1, . . . , ρ ' * — 1) and for \k < i
can also turn out to be nontrivial.
We shall denote ΦΞωΦ by Pk when ω = (ρ - 1, . . . , ρ - 1) (k times).
We denote ΦίΡΙ'Φί by Pk. For ρ = 2 we set Pk = Sqk.
As in the ordinary Steenrod algebra mod p, we have here the following fact: the operations Ρ
together with Λ ρ generate the entire ring Ap
(it suffices to take Pp ). This follows easily from the
fact that for the ring Ap (& Zp it is easily derived from the properties of the ordinary Steenrod
algebra. Hence, it suffices to determine only the action of the operators Ρ
(and
p
even only of the P
on the generators yi
).
We now consider the ring Dt, operating on the module Qp L y J , and the groups E x t p f i ( ^ p L y J ,
Qp [ y j ) . We set
We consider the groups r s > t <8> Λ ρ ; [ Μ . We have
Lemma 1 1 . 2 . a ) There
is a well-defined
graded
action
of the ring Aptl.1
on S r s '
J
®ΛΡΐέ.,
such
that:
1 ) λ ( % ® μ ) =Λ : ® λ μ , λ , μ € Λ ρ
2) if eitW
> ί Μ
C Ap>i.,-
=Φ£5ωΦΙ,
e i | ( u e Ap, i_i, ω =
(pi — I,. . . ,p>* — 1 ) , / ; < i,
then
e*u>(x® μ ) = 2
β*ω(χ)®β1ωζ(μ),
ω=(ω,,ω2)
here
eifO)
( Λ ρ > j . j ) is the ordinary
action
and e j ^ ^ x )
6 Vs (8> Λ ρ , ; - ι for χ €. Vs , μ 6
A
P i i
.l;
b) we have the equality
Horn A* p .^(Îp.i-i, Γ··«®Λρ, ( _ι)
=
HomA j
Proof. P a r t (b) i s obvious.
i i M
(AP,i-i,AP,t^)
T o c o n s t r u c t the a c t i o n of.4p
j . j o n Γ 5 ® A p > 1 . , w e n o t e t h a t t h e ring
β ; a c t s o n Λ ρ > ; = Qp [γ] ® A p > 1 . 1 n a t u r a l l y , w h i l e t h e a c t i o n i s t r i v i a l ο η Λ ρ i_i. F r o m t h i s f o l l o w s t h e
n a t u r a l a c t i o n of t h e f a c t o r - r i n g APti.l
Ext D . (AP,ir
on the groups
Qp [Vi]) = E x t D . (Qp [y], QP [y]) ® Λρ.ί-ι,
w h e r e Dt = Qp [yi · &i. It i s n o w e a s y t o d e r i v e p a r t ( a ) .
884
S. P. NOVIKOV
We note that the ring Β ι is a free right module over &;.
Theorem 11.1. There exists
a) Ea> is associated
b) Ev'q
coincides
a spectral sequence (Er, dr), where:
with Ext/ι
(Λρ,,-,, Γ" ® A ^ . J , where Γ" <g> Λ ρ . ^ is a Αρ^.,-module
ρ
virtue
(ΛΡ;ί, APji);
with Extp.
by
i• ι
of Lemma 11.2;
c) dr:Epr'q
-> £ P + Γ · 9 * r + ' ; - a // differentials
dr preserve the dimension of elements induced by the
dimension of rings and modules;
d) E*-° = ExtpAp^{KPti.1,
KPti.x);
e ) iAe spectral sequence (Εη
dr) is a spectral sequence of rings, where the multiplicative
is induced by the diagonal Δ of the ring
structure
Apii.
The proof of this theorem is more or less standard and is constructed by starting from the double
complex corresponding to the central extension Βi of the rings fcbt, APti.t
For what follows it will be useful to us to compute Ext^'
. We shall not give it here.
(Qp [γ], Qp lyi).
We note that
APli=1Dl,
and the calculation of these groups gives certain information about the ring
ExtAu (U*(P),
Lemma 11.3. Let C be a bigraded differential
U*{P))®ZQP.
ring over Qp, which is associative
and is generated
by elements
such that:
Λ. ι
7//tjj_i —— We. ·*ΐ|
2)
" • CCfbj —~ ftjCCl
d(x)=phi;
3) d(hi) — O,
d(hj+l)=
2 (
,
)hj+i-k-hh,
j ^
4) d(uv) = (du)v + (— 1)έϋ(ο?ι>) where u € C1' . Here, d is the differential
Then the cohomology ring H
(C) is canonically
1;
in the ring C
isomorphic to the ring Ext/^. (Qp LyyJ, Qp
Υγρ).
The proof of the lemma consists in constructing a D;-free acyclic resolution F of the module
Qp L/i + i-l having the form Qp [y] · F, where F is a standard resolution over fe; of the trivial module
Qp = &i/&i, where &i is the set of elements of positive dimension and &t is described in Lemma 11.1.
The ring <Sj has a diagonal, as do D; and Qp LyJ. Hence the complex Horn*)* (F, (? p LyJ) is a ring,
which coincides exactly, as is easy to verify, with the ring C together with the differential operator d.
Whence the lemma follows.
From Lemma 11.3 it is easy to derive
Lemma 11.4. a) For ρ = 2 the cohomology ring Η
bra Z 2 [χ] ® Z 2 [hl, h2, . . . , h2k) with Boks'tem
2) ρ (h2k) = /i2k-i, « > 1, x c. Η ·
(C ®Z2) is isomorphic to the polynomial alge-
homomorphism β of the following form:
, h2k t H2
<2 -1>(L *& Δ2).
b) For ρ > 2 the ring Η
885
(C ® Z p ) is isomorphic to the ring
Zp[x]®A[hu
hPi...,
hph,...]®Zp(y2,
...,yk,...),
where
and the BoksteXn hornomorphism β has the following:
p
c) The group Exti^t(Qp
L y J , Qp L y J ) is nontrivial for t = 2 p ( p ' — 2), </ > 1, ara</ is isomorphic
to
the cyclic group Zf(q), where f(q) — 1 is equal to the largest power of ρ which divides q. We shall
denote the generator of the group E x t p ' 2 < ? ( p I ' l ) (Qp [y], Qp [ y ] ) by δ , .
d) The image of the homomorphism of "reduction modulo p,"
aP
is generated by the following
elements:
9 1
1) AjX " for p> 2 and all q,
2) h^'1
for q = 2 an<i </ = 1 mod 2,
9 1
3) Α , χ ' + A 2 x ? ' 2 for ρ = 2 and q = 0 mod 2.
e) For a/Z f > 1, in the groups H'' * (C; <S> Z 2 ) i/ie kernel Ker /3 coincides
with the image Im
Hence, the homomorphism of reduction modulo p,
aP : Extc* «? p [y,L <?P [tfi])-»-^·· (C< ® Z p )
i's an isomorphism on the kernel Ket β = Im/3 anci none of these groups has elements of order p2.
The proof of (a) and (b) follows easily from the form of the ring C — in particular, from the fact that
C ® Z p is commutative, C is obtained from the standard ©(-resolution and G>; <S> Zp has a system of
generators [I
;
i , while
ff" (Ci (g) Z p ) = Z p [x] (g) E x t ^ , Θ Ζ ρ (Z p , Zp).
The structure of the Bokstem homomorphism β is derived immediately from Lemma 11.3.
Part (c) follows from the fact that e%,h{Vi ) == 2j {
)pkxq~h,
as was shown in Lemma 11.1,
and from the construction of the standard fe;-resolution F for the module ©;/δ; = Qp and the differential d* in the complex Horn* (Qp [ y j · F, <?p t y j ) . Namely, we have:
886
S. P. NOVIKOV
1
q
-j^jd(x )
Part (d) is derived from the fact that
1
q 2
and is equal to h^'
+. h2x '
9 1
mod ρ is equal to Λ,χ ' for ρ > 2 or ρ = 2, q = 2s + 1,
for ρ = 2, q = 2s.
We shall now prove part (e). Since the homomorphism β is a differential operator, it suffices to
l
show that H (H (Ci ® Z p ) , β) = 0 for t > 1. The structure of the homomorphism β was determined in
parts (a) and (b) of Lemma 11.4, and the required fact is easily derived from the usual homological
arguments. The lemma is proved.
1. The ring structure in Ext*D*(qp Ly], Qp LyJ) completely follows from Lemma 11.4, since the
homomorphism of reduction modulo p,
dp : E x t " . (Qp[y], QP [y])-> H** (C< ® Zp jt
is a monomorphism on Ker β and in dimensions > 2; hence, from CLP (χγ) = 0 it follows that xy = 0 for
elements χ, γ of positive dimension. The image of the homomorphism a p (ExtJ*) coincides with Ker β
in all dimensions > 1, although Ker ap is nontrivial in dimension 1 Lsee parts (c) and (d)J.
2. The product Ext^'* ® Ext^' * (Qp [ y ] , Qp [ y ] ) is identically equal to zero for ρ > 2.
3. A basis for the group Ext^* (Qp [ y ] , Qp [ y ] ) is completely given by the set of elements:
a
) αΓ } = β (hhxm),
b) ah
= $(h2kxm)
m ^ 0 .where ρ > 2, where
kîx
= (h2k-ixm
+ mhzkhiX"1^)
where ρ = 2, k > 2, m ^
0.
4. For ρ = 2 the product Ext^' * Θ Ext^* — Ext^J* is defined by the formulas:
a) 02?+i · θ2ί+ι — αϊ
b)
O2q+l-O2l =
,
c)
χ
,
αϊ
,
(2q+2l-2b
χ
02z · o 2 m = αϊ
(2q+2l-i)
+ α2
1 2
In particular, we shall denote the element S t by h 6 Ext ' . Hence, from (a) and (b) it follows that
025+1 · <%n = /i02g+m
We note that Dl =ΛΡΛ
for all
q, m.
and there is defined a natural homomorphism
yd)
#
Ext D ; (Qp fal Qp [yi})-+
E x t ^ { V (P), U* (P)) ®z Qp.
From Lemma 11.4 and the results of s §7, 8 is derived the following
Theorem 11.2. a) For t = 1 the homomorphism γρ
is a monomorphism.
1
b) For all ρ > 2 the homomorphism γρ ^ is an isomorphism.
c) For ρ = 2 the homomorphism γρ
is an isomorphism on the groups E x t 1 ' 2 ' for q = 2 and for q
odd; for q = 2s, s > 2, the image of the homomorphism y^
lias index I or 2 in Extl'2J
®zQi>
an
d
in
fact index 2 for all q = As, s > 1.
d) For all q = 4s + 1 and q = 4s + 2 the image Ιΐαγ[1^ coincides
variant (jrp (7r»(S™)) = q\? (/π* (SO)). For ρ > 2 the image Imy
invariant
qU (π* (Sn)) = q[' (In* (SO)) in all
dimensions.
with the image of the Hopf in-
coincides
with the image of the Hopf
In the formulation of Theorem 11.2 the calculation of the group Ext 1 '^
is the homomorphism y 2
+ 4
®zQi
887
l s
n o t
complete -
an epimorphism or does Imy 2 have index 2?
For the study of this question we shall use the spectral sequence (Er, dr), described in Theorem
11.1, which converges to the groups Ext^ 2 (Λ 2 > 2 , Λ 2 2 ) . Namely, we must compute the groups E^'1
and the differential
Λ
d2 : Ε2° -+ΕΪ'° « Eiti, ;i (A2,,, Λ 3ι1 ) = E r £ , (&foi],
The groups Ext/) were computed in Lemma 11.4 for all ρ > 2. We may assume that y, = [CPp'll
€.
and
where χ ι = [CPP " ' ] . Moreover, by the integrality of the Todd genus we can set λ = ρ — 1 and
y2 = ~ (x2 + (p— i)x£+i)L
yi = Xi-
Ρ
We have:
•AP.i = Qp[yi],
APtz
T h e a c t i o n of t h e o p e r a t i o n Φ · Ρ · Φ o n Λ
ρ ι
=
Qp[yi,y2].
and Λ ρ > 2 i s g i v e n by the formulas:
(0, k^p,
Φ-Ρ''-Φ{χ2) = Λ ρ ) Χ ι '
p + l,
k==
P'
As a consequence of this we have
Lemma 11.5. The action of the operators
following
Ρ on the generators
y 2 of the ring Λ ρ > 2 is given by the
formula:
where (l2, . . . , l^) is an ordered partition
of the integer k and
a) P<(y2)= j(P*(x2) + (p - l)F(xf + 1 )) = (Ρ-ΐ){ρ+ιί)ρΐχι+ι-1
b)
for ι φ
Pt
S. P. NOVIKOV
l
p
We note that P (y 2 ) is divisible by ρ for 1 ^ ρ and P (y 2 ) is not divisible by p.
Now we can describe the action of the ring Ap>l
Qp [ y 2 ] ) and Λ ρ > ι = Qp [ y j = Qp [XJ,
P
part (c). The generator of the group E x t '
?(p2
s
= Ext^ ' (Qp [ y 2 ] ,
1
The groups Γ '* were computed in Lemma 11.4,
= [CP ~ ].
Xl
1
ι ε
on Γ ' ® Λ ρ ι , where Γ ·
1
l)
" ( ( ? p [y2], Qp iy2})
q
flq)
is obtained as d* (x )/p
in the
complex Honip (F, Qp [ y 2 ] ) where F is a D2-ftee acyclic resolution of the module Qp [ y 2 ] ,
9
x € Horn* (D2, Qp [ y 2 ] ) is an element such that %9 (1) = y2 , f(q) — 1 is the maximal power of ρ which
divides q, and d is the differential in the complex Horn! (F, Qp Ly 2 J).
We set
where
by virtue of Lemma 11.5 and ak € <^p Ly,J, α^ = λ yf 4 .
Lemma 11.6. The action of the ring APil
From what has been said it is easy to derive
1
= Di on Γ ® Qp [y 2 ] is described in the following
fashion:
9-1
are generators (their orders are ρ
) and a^ €. Qp iyji
is described in Lemma 11.4.
Lemma 1 1 . 6 f o l l o w s e a s i l y from Lemma 11.4 and the d e f i n i t i o n of the generators aq = d xq /ρ
where xq C Horn* (F, Qp Ly 2 J) i s s u c h that xq (1) = y2 G Qp L y 2 J . Further, w e compute Horn*
Γ 1 ® Λ ρ 1 ) = El'1 in the s p e c t r a l s e q u e n c e (E2, d2) of Theorem 1 1 . 1 , which c o n v e r g e s to E x t ^ *
Λ ρ > 2 ) ; here ΑρΛ
= D1 and Λ ρ > 1 = Qp
,
(APjl,
(Λρ>2,
[yj.
L e m m a 1 1 . 7 . The groups
2
Horn*-
(ΑΡι1,
Γ
2 p ' ( p — 1 ) + 2q ( p — 1 ) for all i > 0 , q > 0 , where
1
®Λ
ρ ι
) are spanned
the order
by generators
of the generator
K ; J 9 of
dimension
K j > q is p .
The proof of the lemma follows easily from Lemma 11.5 and 11.6 by direct calculation.
Since Horn*
(Λ ρ > 1 , Γ'(8).Λ ρ > ι ) = /s"' 1 , our problem is to calculate d2 r i " ' 1 -^ E2'° = Ext^
(APjl,
Λ ρ > 1 ), where the latter groups are computed in Lemma 11.4 and in the conclusions drawn from it.
Direct calculation proves
d2 -.El'1 -> E22'° of the spectral
Lemma 11.8. The differential
Ext4
(^p,2> ^p,2)
i s
sequence
889
(Er, dr) converging to
given by the following formula:
where h ι and χ are in the notation of Lemma 11.4, and β is the Bokstem
— Exto (Qp C*J) described
homomorphism Η (C)
in Lemma 11.4.
From Lemma 11.8 follows the important
Corollary 11.1. a) For ρ > 2, the kernel Kerd2\E°2l is trivial;
b) For ρ = 2, the kernel Ketd2\E2'1
is generated by elements
κο,2ί+ι e= H O I U A 4 ^ (Λ Ρ ι ι; Γ 1 ® Λ Ρ ΐ 1 ) .
* > 0.
Hence, the image of the homomorphism
,ι; A P i l ) - ) - E x t A p | 2 (Λ ρ , 2 ; Λ ρ , 2 )
;)il
has index 2 for all ί > 0.
Parts (a) and (b) of the corollary are derived in an obvious way from the structure of the homomorphism β, which was completely described in Lemma 11.4. The sharp distinction between the
cases ρ = 2 and ρ > 2 is explained by the fact that for ρ > 2 we have h\ = 0 and β (hlxs)
+
s > 0, while for ρ = 2, β (h.x23
= 0 for all
' ) + 0.
Comparing part (b) of Corollary 11.1 with Theorem 11.2, we obtain the following result.
t ^ 4, the order of the cyclic group Ext1' ^ (U* (P), U* (P))
Theorem 11.3. a) In all dimensions
coincides
exactly with the order of the group E x t l ' l (Κ (Ρ), Κ (Ρ)), and this isomorphism is induced
ψ
by the Riemann-Roch functor λ.
b) The Hopf invariant
πη+,-ι^—ΕχΙ 1 ·' (*/·(/>), V (Ρ))
:
qi
A
is an epimorphism for t = 8k, t = 8k + 2 and t = 4, and the image lmqi has index 2 in Ext 1 '' for
t = 8A: + 6, k > 0, and I = 8k + 4, k > 1.
Corollary 11.2. The generators aq of the groups E x t 1 ' ! ' (U (P), U (P)) are cycles for all Adams
differentials
dt for q = 4s, 4s + 1, s > 0, and q = 2, and are not cycles for all differentials for
q = As — 1, 4s + 2, s > 1 (the elements
Supplementary
remark.
2aq are cycles
for all
It i s p o s s i b l e in a l l d i m e n s i o n s
differentials).
t o p r o v e t h e f o r m u l a di{a.q)
= A 3 · CLQ.2
for q = As — 1, 4s + 2, s > 1, where h = al €. Extt 1' '^. In particular, for (/ = 4s + 2 this follows from
1
i
the fact that h ciq.2
4
φ. 0 in Ext y, while at the same
e time o.q_2 is realized by the image of the J-
homomorphism, and we must have in £ „ that Α 3 α ς . 2 = 0.
*We take this opportunity to note the small computational error in parts (3) and (4) of Theorem 5 of the
author's paper L
J, which is completely corrected in Theorem 11.3 and Corollary 11.2 of the present paper.
890
S. P. NOVIKOV
§12. The Adams spectral sequence and double complexes.
Comparison of different cohomology theories
We assume that there is given a complex Υ = Υ .ι 6 S and a filtration
Y^-Y0^-Yi^...^Yi^...,
where the complex of /l*-modules \X* (Yif
M=
Yi+ 1) = ;1/;1
{Μ05-Μ1+-Μ2-<
ί-Mi*
}
is acyclic in the sense that Ht(M) = 0, i > 0, and H0(M) = X* (Y). The modules I/; are not assumed
to be projective. In the usual way a double complex of A
I
l '
I
| dx
-free modules Ν = (/V£/) is constructed.
Jd, j d ,
····
No,o£-NOti£.No,*+-
such that (a) <^ο?2 = - rf2rf,; (b) ί-> . . . - /V[; Ίΐ /V£.l>;· -> . . . I for all / i s an ^ X - f r e e acyclic resolution
of the module W;-; (c) if ζ>^ = 2 ^ » 3
a n d
^ = î + ^2: Qk — Qk-i, then the complex \Qk ^ Qk-i^ • · · I
is an -4 -free acyclic resolution of the module X* (Y); (d) the complex /V; = S— Nitj
->...}
-^Nifj.l
is such that Hk (/V,·) = 0 for k > 0, //O(/Vj) i s a free / ^ - m o d u l e and the complex j . . . Ha(Nk)
d
-i
H0(Nk-i
- · . . . ! represents an A -free acyclic resolution of the module X (Y).
As usual, there a r i s e s a spectral sequence of the double complex (Etr'q,
dr), where
and
with L an arbitrary A -module; this spectral sequence converges to Ext
χ
(Χ (Υ), L).
Definition 12.1. By a geometric realization of the double complex V = (;Vi;) in the category S or
S (8> zQp
1 S
meant a s e t of objects ( Z i ; ) , i > — l , / > — 1, and morphisms
Z_! _ ! <— Z_!_ 0 <— Z _ l jx <— · • ·
I
ZQ
1
_X <—Z,, 0
i
<—ZOii
ί
\
ί
t
ί
ί
<— • · ·
with the following properties:
a) Ζ-,,-i = y, y^ = Z.1j and the filtration Z . l j . 1 <- Z . l j 0 <- . . . . coincides with the filtration
y -
y0 - ' . . . -
y
;
- ..."
891
b) The filtration
Yi I Y%+l — Z~i, i I Z—i., t+i "*— Zo, i I Zo, i+i -*— Zit i I Z\, i+i - < — . . .
represents a geometric realization of the A -free resolution of the module X (Y c/Yi+ t) = Mi,
a n d h e n c e A * ( Z f t > £ / Z j . j i + 1 IJ Zk+lii)
c) T h e differentials
d1 :Nkti
=
Nkii.
-· !^k-i,i
a f i
d d2 :Nkti
— i^k,i-i
coincide with the natural homomorphisms
di
X* {Zk,i/Zh,i+l [} Zh+l;i)—*~'X* (Zh-l,i/Zk,i
[} Z^—l, i+l) ,
d2
X* {Zh,i/Zk,i+l
U Zk+l,i)^>~ X*
(Zk,i-l/Zk
We make s o m e d e d u c t i o n s from t h e p r o p e r t i e s of t h e g e o m e t r i c r e a l i z a t i o n of a d o u b l e c o m p l e x :
1. T h e filtration Z . 1 ( . , = Υ <- Z oM «- Zly.1
< - . . . < - Z ; _t <- . . . r e p r e s e n t s t h e g e o m e t r i c r e a l i -
z a t i o n of t h e / l X - f r e e r e s o l u t i o n \H* (V o ) ί2 Η* (,Υ,) «- . . . Κ
2. T h e filtration Y «- Z.t
| J Z O.,<-...
Β
U
Z j ; *- . . . r e p r e s e n t s the g e o m e t r i c r e a l i z a t i o n
i+j=ft-i
of t h e A -free
resolution
where d = dl + d2.
3. T h e d o u b l e c o m p l e x ( Z ) d e f i n e s two A d a m s s p e c t r a l s e q u e n c e s :
a) t h e A d a m s s p e c t r a l s e q u e n c e ΕΓχ
in the theory X , i n d u c e d b y t h e filtration
y "*~ ^ o , - i U •Z-i.o -«—...-«—
U
Ζ,, j -«—...
with term £ * = Ext 4 ^ (A"* (Y), A* (K)) for any /( € S;
b) the spectral sequence Er of the filtration
with term £, = iHom* (K, Yi/Yi
+ i)\.
In view of the presence of the double filtration (Ζί;·) of the complex Υ in all terms of both Adams
spectral sequences there arises yet another filtration: in the first case it is equal to φ(χ), x € E^,
where φ(χ) coincides in E2 with the filtration in Ext* χ (Χ (Υ), Χ (Κ)) induced by the non-free
resolution X (Y) *~ Mo — Mx •- . . . , and in Ε^ is induced by the geometric filtration
U
i+j=fe-l
Zi,} = 5 . . . =>
U . Zu
i+j=h-l
= . . . = > Z_ilft.
c) For the second Adams spectral sequence the filtration in Er and £co is induced by the geometric filtration
892
S. P. NOVIKOV
We shall denote it by Ψ(7), y € Er.
In addition, each of the indicated spectral sequences defines in the groups of homotopy classes
of mappings Horn (K, Y) the usual filtration i(x), whose corresponding index i is such that the element χ £ Horn* {K, Y) is nontrivial in E1^ and trivial in E'^ for / > i. For the Adams spectral sequence
of the theory X we shall denote this filtration by ίχ. We have the double filtration ιίχ (x), φ(χ)],
where χ £ Horn* (Κ, Υ), ψ (χ) < ix (χ).
The second Adams spectral sequence for Horn (Κ, Υ), induced by the filtration
also induces a double filtration in Horn* (K, Y): li(x), Ψ(χ)1.
From the construction of the double complex it is obvious that we have
Lemma 12.1. The filtrations described above are related by
i (Χ) 5ξ φ (X) ^ ίχ (Χ) 5ΞΞ ί (Χ) + Ψ (Χ)
for all χ €. Horn (Κ, Υ) in the presence of a geometric realization of the double complex defining both
Adams spectral sequences.
By standard methods one proves
Lemma 12.2. If X is the theory of Zp-cohomology, then for any acyclic filtration Υ = Yml <-' Yo
<- Yl <- . . . there exists a geometrically realizable Α-free double complex (Z), where A is the ordinary
Steenrod algebra.
The proof of this lemma is obtained easily by the methods of L ] .
The most important example which we consider here is the theory of cobordism in the category
s®zQP-a) Y G Dp, i.e., Η {Υ, Qp) has no torsion.
b)X
= H*( ,Zp),Ax
=A .
c) The filtration 7 3 Yo D Y, 3 . . . is an acyclic free filtration in the theory U ® zQp o r in the
theory Up C U ® zQp· By virtue of the exactness of the functor Up -> Η ( , Zpp)) in the category Dp,
the filtration Υ D Yo 3 . . . is also acyclic (although not free) in the theory X = Η ( , Z p ) .
In this example, the filtration i(x) is a homotopy invariant, with i(x) = i,t (x), where Up is
cobordism theory. Moreover, we have
Lemma 12.3- a) All the filtrations ίχ, iy , φ, Ψ, for X = II { , Z p ) , a Up-free acyclic filtration
Υ 3 Yo D Y, I) . . . in the category S ® zQp an<^ anJ X -free acyclic double complex (Z) are homotopy
invariants of Υ, and we have the following inequalities:
iu*p (x) ^ φ(ζ) «ξ ΪΗ*( , ζρ)(Χ) < iu*p (x)
where i = irj*, ίχ = L t ,
ζ
+ΎΡ(Χ),
. are the respective filtrations in the theories U 0 zQp and Η ( , Z p ) .
b) The second Adams spectral sequence Er coincides in this case with the Adams spectral
sequence in the theory U <8> zQΡ for
r
> 2.
c) Both Adams spectral sequences Er in the theories X = II ( , Z p ) and U ® zQp (or Up) ί η
our case preserve, respectively, the filtrations φ and Φ.
893
d) The Adams spectral sequence in the theory [I ( , Z p ) is such that each differential dr for
r > 2 raises the filtration φ at least by 1, i.e..
For
the proof of (a) we note that the Up-filtration
Υ 3 }' o 3 } \ D . . . depends functorially on the Up-
free resolution and is uniquely determined by it. For a fixed Up-filtration
the same thing is true with
respect to the double complex Ν and the double filtration (Z) defined by it. Parts (b) and (c) are
obvious. Part (d) follows immediately from the fact that the complex Υ JYi
+l
is a direct sum of
spectra Wp of the theory Up up to suspension. For such objects the Adams spectral sequence has
zero differentials for r > 2, as was proved by Milnor and the author L
'
J.
The lemma i s proved.
We now consider the graded ring Λ ρ C Ωυ ® zQp, where Λ ρ = Qp [xu . . . , % , , . . . ] , dim*! = 2 p ' - 2.
The ring Λ ρ is a local ring: it has a unique maximal ideal m C Λ ρ such that Ap/m = Z p . Hence the
bigraded ring Λ ρ = ^j mt/mi+i
is an algebra over Z p , and Λ ρ = Z p [h0, h^ . . ., h^ . . . ] , where h0 is
ρ
<—0
a s s o c i a t e d with m u l t i p l i c a t i o n by ρ and dim A; = ( 1 , 2 p [ — 1), i . e . , hi £ m/m2. C l e a r l y f]ml = 0 and,
by
[15'17'
1 8
] , we h a v e :
Ap = ExU
(H'(Mp,Zp),Zp).
As was established in s 11, the action of the ring Ap
on Λ ρ = Up (P) preserves the filtration gener-
ated by the maximal ideal m. Hence it defines an action on Λ ρ , which is described as follows:
1) the action of A p o n A p is defined by multiplication;
2) the action of Ρ
on Λ ρ is defined so that
Pi>'(/n)= hi-i and Pi(ha) = 0T
/ 5=s 1,
Pk(ab) ~
^
Pl(a)P*(b).
l+s=k
We c o n s i d e r the ring A a s s o c i a t e d to l p by the filtration Ap D mAp D . . . Γ.· miAp
?)....
We note
that in the ordinary Steenrod algebra A there i s a normal (exterior) subalgebra Q C .1, Q = A(Q0, . . .,
Qi,
. . . ) , d i m ^ t = 2p' - 1, s u c h that A/./Q i s isomorphic to the quotient Α/βΛ [J Αβ and Ext/i (//* (Mp,
Zp)) =ExtQ(Zp,
Z p ) =A
p
=Z
p
[k0, h
l ?
...,
hi...A.
From the results of § 11 and the structure of the Steenrod algebra .1 follows
Lemma 12.4. The algebra A associated to the ring Ap
is isomorphic to (Ap · A//Q)
, where the
commutation law ah = %a* (h)al is given by the action of A//Q on Ap defined by the formulas
Ppr(hr)
= /t rM , r > 1, Ph (h0) = 0 for k > 0, and Aa = Σα ; ® 5 ; , where A: A//Q -. A//Q ® A/./Q is the
diagonal and Ρ
is the ordinary Steenrod power.
We note now the following identity:
t / (Λ ρ , Λ ρ ') = Extiz/Q (Zp,
Aj) = Exti / / Q (Zp,
(here, t is the dimension in Ap defined by the filtration Ap = mt/'mt
L
p
= <.'*(}•) and Μ = Η* (Υ, Ζ ρ) = L/mL we have:
a) 1/ is an /l//D-module;
+l
Ext Q f (Zp, Zp)
) . Moreover, if } C Dp, then for
894
S. P. NOVIKOV
b) there exists the identity
V ) = EXI'AJ/Q (M, Ext Q J (Z p , Zp)),
Extj'(L,
where L = S/re'L/ra' + 'L is an A -module and, clearly, a Λ -free module.
~
Pa
Two spectral sequences (Er), (Er) arise, both with the term
M, Ext Q (Z p ,Z p )).
These sequences have the following properties:
1) In the first, which converges to Ext^ (M, Z p ) , we have
2) In the second, which is induced by the filtrations in Λ ρ , Ap , L and which converges to
Ext . υ (L, Λ_), we have:
Λ
ρ
3) d, = dl and £/•' = £ / · ' = Εχι* / / ρ (i/, Ext^ ( Z p , Z p ) ) .
4) In both spectral sequences there is yet another grading Ep = ^)j Ey ' q and Er' ' = ^ j Esr' '9',
q
q
induced by the dimensions in all modules and algebras which appear, and connected to the spectral
sequences as follows:
a) the third grading q is preserved by all differentials dr of the spectral sequence Er which converges to Ext/i (M, Z p ) ;
b) since
^/j
Λρ
is associated to Λ ρ ', the third grading q in the second spectral sequence
t—q=m
ET, which converges to Ext y (L, Λ ρ ) , is increased by r — 1 by the differential dr:
A
p
r, t-τ+Ι, q
y
~ s,t',q
2 &co
s+t=m
b) The group
2ll
ό 'Q
-^οό ' Q
l s
. 75 s , ' , 1
P s +L. i+r-1, q+r-l
is associated with
associated with
ExtTup (L, Λ Ρ ) = Extfu (U* (Y), U'(P)) ®zQPt
where L = U*p(Y),Ap = U*(P).
Thus, in the groups
ttlQ = Ext2J/Q(M, ExtQ(ZP, Zp))
895
we have two "dimensions": (m, q) = (s - !, q) is the "cohomological" and (s, q — t) = (s, /) i s the
"unitary" (in 6-cobordism). The "geometric" dimension (of the homotopy groups) is equal to q — m
- I — s = q — s — t.
We note the important fact: the dimension of the element dr(y) for the element γ of "unitary"
dimension (s, /) is equal to (s + r, I + r — 1), where / - q — t; and, conversely, dT (y) of an element
of "cohomological" dimension (m, q) has "cohomological" dimension (m + r, q + r — 1), m = s +- t.
This means that both these spectral sequences have the form of the Adams spectral sequence,
although they are defined purely algebraically by the ring 4 p .
Up to this point there has been no difference between ρ = 2 and ρ > 2, if we speak of the results
of this section. However, the following theorem shows the comparative simplicity of the case ρ > 2.
Theorem 12.1. For any ρ > 2 and complex Υ C D p, the spectral sequence (Er, dr) has all
differentials
dr - 0 for r > 2. The groups
s+i=m
are
s+t=m
to Ext™' g (;l/, Z p ) , where Μ == Η* (Υ, Ζρ),
isomorphic
( Z p , Zp) = Zp [h0, ...,hu...l
and
way:
the algebra A//Q generated
Ppi(hl
)=hh
+i
Pk(hO)
d i mfti= (1, 2p* - 1 ) ,
by the Steenrod powers Pp
acts on ExtQ ( Z p , Z p ) in the
following
2
= 0 fork>0,and
From Theorem 12.1 follows
Corollary 12.1. For any complex Υ £ Dp? where ρ > 2, there is defined an "algebraic
spectral
sequence"
(ΕΓ dr),where
is a s s o c i a t e d t o E x t ™ ' 9 (1/, Z p ) , d
sociated
to Exts ·}, (UZtY),
Es2-l-q
T
: E
= E x t ^ g (,tf, ExtQ(Z p , Z p ) ) , iAe group
s
q
T
-> Ef
+
+ r-i, ? + r-i^
an(
^ the group
•#!' ? ' ' = E?'
2
2.
Adams
^ c a '7
q
is as-
Μ = //*(Y, Z D ) .
Ut(P)),
4
• P
We prove Theorem 12.1.
the
In the Steenrod algebra 4 for ρ > 2 there i s defined a s e c o n d grading —
s o - c a l l e d "type in the s e n s e of Cartan," equal to the number of o c c u r r e n c e s of the homomorphism
β in the iteration. We s h a l l denote by τ (a) > 0 the type of the operation a £ A, with
4
2 ^ττ
=
τ
T
T
T
where τ is the type and A » · <4 2 C 4 i
+T2
· By the same token, for any Υ Ε Dp there is an extra
grading — the type r — in the groups Ext/i (M, Z p ) , and
ϊ
(M,ZP)=
where / — r s 0 mod 2p — 2. We notee that for
for Q C
C 1, r(() r ) = 1 and r(Ph) = 0. It is also obvious that
r(hi)
= 1, Λ, € ExtV, (Z p , Z 2 ) , and the type is an invariant of the spectral sequence (dr, Er) for r > 1.
896
S. P. NOVIKOV
Since the type is trivial on the ring A//Q, and A//Q C A, all dT - 0 for r > 2, since on the groups
s
Ext A'^Q ( Z p , Ext^ ( Z p , Z p )) the type r = ί and r(dry) = r(y) for r > 1.
This implies the isomorphism
q
2
ExC (M,Zp)=
Ext^ / Q (M,Extp(Z P l Z p ))
and Z?2 = Em. The theorem is proved.
From the proof of Theorem 12.1 follows
Corollaryl2.2. The second term of the "algebraic Adams spectral sequence"
1 11
l q
12.1 is canonically isomorphic to the sum Έ,Ε^' ' ,
type, Μ = Η* {Υ, Z p ) for Υ € Dp, and
In this spectral
E x t f ' 9 (Λ/, Z p ) = Ext™'
2
9
(Er, dr) of Corollary
(M, Z p ) , t is the Cartan
(M,ZP).
sequence
Ύ
and the group
= Ext^'
where El' '
il<7
E^''9
2
i s
. J7 S , t, q
associated
'j7 s + 1 > t+r-1, 7 + r - l
to Ext 5 '' (U* (Y), Up(P)).
A
t~q=l
Ρ
From the geometric realization of double complexes as defined above, Theorem 12.1 and Corollaries 12.1, 12.2, there follows
Theorem 12.2. The "algebraic Adams spectral sequence"
spectral sequence (Er, dr) in Η ( , Zp)-cohomology
1) ET-q = 2
to the Adams
theory for all ρ > 2 in the following
sense:
1|· '• = Εχί%·9(Μ, Ζρ);
2) if for some γ €. El'l'q
φ(γ
(Er, dr) is associated
we have (/; (y) = 0 for i < k and dk(y) /- 0, then there is a y such that
— y) > φ(γ) + 1, di(y) = 0 for i < k, and dk(y) /- 0, and moreover φ (d^y) = φ (y) + 1, where
Φ(y ) ~ Φ ()') — ^ β/ϊα φ(ά]ίγ
— d^γ) > φ(ν) -t 1;
9
3) t'/y'e Ε χ $ ' ( Μ , Z p ) fs suc/t i/iat ^(y") = 0 for i < k and φ(ά^γ) > φ(γ) + I, then for the projection y of the element γ in Ext™
note that for elements J £ 2 J
y
<y
' q [M, Zp) we have the equation di(y) = 0 for i < k (we
Ext^'' 9 (Μ, Ζρ), φ (y) ^> a ) .
The groups E x t 1 ' ! (Up (P), Up (P)) were computed in previous sections; they are cyclic for
Ά ρ
s = 2k (ρ — 1) of order P'* ', where f(k) — 1 is the exponent of the greatest power of ρ which divides A;.
Corollary 12.3. The generator ak of the group Ext 1 '!*
in other words, φ(α/ί)
" O ( ' V <%) has filtration (1, k - f(Jc))>ror,
= k — f(k) in the term Ew of the 'algebraic
for ρ > 2. Since Ext'*u (Λ ρ , Λ ρ ) consists
i > 2, and there is an associated
< k - f(k) + 1.
( i
Adams spectral sequence"
of cycles for all Adams differentials
(Er, dr)
in Up-theory, dt (a/c) = 0,
element ak Ε nt (Sn); we have φ(αιί) = k — f(k), iji (a^-j _ 1, i^ (a^.)
P r o o f . A s w a s s h o w n in ^ 1 1 , t h e h o m o m o r p h i s m E x t '
( A p i , A p > 1 ) — E x t 1 y ( A p , A p ) i s an e p i -
m o r p h i s m for ρ > 2. F o r t h e ring Dr and t h e m o d u l e A p i l — Qp
d e t e r m i n e d ( s e e L e m m a 11.4), w h e r e φ(χ)
897
t h e ring C -- Horn (l· , A p l )
lxlJ
-•• 1, φ (hi) = 0, φ (ρ) ~- 1 a n d d(xh)^=^
Σ f
\ / /
was
)
)d(xk).
T h e e l e m e n t a.^ w a s r e p r e s e n t e d by a j . = (l/p''
F r o m t h i s we h a v e :
•
·Γ / * \ , ·
,
1
cp(aft) = ininy φ( . )+J + k —j — f(k) \ = k — f(k)
x
I \ j !
J
ι
Thus, the filtration φ of the element ak is equal to k — f(k), since the filtration φ is induced by the
filtration in the ring A p . The Corollary is proved.
As is known, the groups E x t ^ ' s ( Z p , Z p ) are equal to Z p for s = 1 or s = 2p' (p — 1) and are generated by elements it-, / > 0, of type 0 for s - 2p' (p — 1) and h0 £ Ext 0 » 1 ' 1 of type 1 in the sense of
Cartan.
Hence, u, £ E x t ^ 0 ' 2 " ' ( p - ° (Z p , Z p ) and h0 £ E x t " ' 1 · 1 ^ , , Ζρ), where E x C = 2
Exti'"9
s + t=m
and I is the type. In the groups Ext^' 2p
there are nonzero elements y,, i > 1, having type 0.
Corollary 12.4. In the 'algebraic Adams spectral sequence" we have the equation d2(ui) = hoyl,
for t > 1.
The proof, by analogy with the proof of Corollary 12.3, follows easily from the structure of the
homomorphism β in 11 (C <S>Zp); where β(ηρί) == γ,; for (· > 1 (see Lemma 11.4).
Thus, we see that with the help of the "algebraic Adams spectral sequence" it is not only possible to prove the absence of elements with Hopf-Steenrod invariant 1, but also to compute (ordinary)
Adams differentials by purely algebraic methods which come from the ring .1 .
Conjecture. For ρ > 2 the "algebraic Adams spectral sequence," which converges to Ext ,,(U (P),
U (P))®zQp-> coincides with the "real" Adams spectral sequence, and the homotopy groups of spheres
a r e
"•» ($") ®zQp
associated to Ext L, (U (P), V (P))&zQp- iEquivalently: all differentials dr,
r > 2, are zero in the Adams spectral sequence over Vp.\
We now consider ρ == 2. As was indicated earlier, here there are two spectral sequences (Er, dr)
and (En dr), where E2 = E2 ^ Ext^//Q (M,\2),
Μ = Η* (Χ, Ζ2), and A2 = Ext** (Z2, Z 2 ) is associated
to U2 (P) = A 2 . The sequence (Er, dT) converges to Ext u(i'2 (X), A 2 ) and (Er, dr) converges to
Ex t / 1 (M, Z 2 ) .
By analogy with Theorem 12.2 for ρ > 2, here we have
Theorem 12.3. The differentials dT are associated with the Adams differentials in Cobordism
theory on the group Εm associated with Ext y (f-'2 (X), A2). The differentials dr are associated with
the Adams differentials in Η ( , Z2)-theory on the groups Εm associated with Ext^ (H (X, Z 2 ), Z 2 ),
where, X £ D.
The proof of Theorem 12.3, as of 12.2, follows immediately from the properties of the geometric
realization of the double complex.
898
S. P. NOVIKOV
Thus, for ρ = 2, it is possible to compute the Adams differentials in Η ( , Z2)-theory, starting
from cobordism, and conversely.
Question. Do the algebraic Adams spectral sequences Er and Er define the real Adams spectral
sequences in both theories?
In any case in all examples known to the author all Adams differentials are subsumed under this
scheme.
Example. Let X = MSU. We consider E x t ^ ^ (Μ, Λ), where Μ = Π* (Χ, Z 2 ). We write an A//Qresolution of the module M:
We recall that Μ = F -f- 2jMa,
where F is A//Q-hee and Μω has one generator ϋω for all ω = (4k1
.. . , Aks) and is given by the relations Sq2uw = 0 over A//Q, where dim ιιω = 8Σλ;. Hence one can
assume that C = C (F) + Σί(,ί/ω), where C (F) = F and 0{Μω) has the form:
C (Ma) = (-»-... - i A//Q -+ A//Q -»-... -»- A//Q + Μω),
where ui is a generator of Οι(Μω) and αίίί; = Sq2!*;-!. The action of Sq2 on Λ2 was indicated earlier:
A2 = Zz[h0,...,hi,...],
dimhi
= ( 1 , 2 £+ 1 - 1 ) , i > 0 , w h i l e S ? 2 / ; , = h 0 .
There follows straightforwardly (by direct calculation)
Lemma 12.5. Ext^** 0 {Μω, Λ2) for ω - (0) has a system of multiplicative
AosExt 0 · 1 · 1 ,
χι e Εχί1·0·2,
A4 e Ext 0 · 1 · 2 ' 4 4 - 1 ,
generators:
yÊxt0·3·6
i > 2,
αίΐί/ is given by the relation hoxi = 0.
We note that the dimension of 'Ex.ts't'q in Η ( , Z 2 ) is equal to (s + t, q) and the dimension in
U2 -theory is equal to (s, q — t) (see above).
We now describe the spectral sequences Er ** Ext. and £ r ^ Ext
y
Lemma 12.6. a) ΓΛε spectral sequence {Er, dr) is such that:
d(h)
d ( ) = 3z{hi)=
S3(vi0)
= 0,
and all dr = 0 for r = 3.
b) Γ/ϊβ spectral sequence (Er, dr) is such that:
d*2{Vczi)) = Xihi+2t
i ^ 0,
22(HomA,/<i(F,At))=0>
where F is A//Q-free, dr = 0 for r > 2 (we no£e ί/ίαί ι;ω
Μω, ω = (ku . . . , ks),dimua>
= 8(ΣΑ;·) and νωνω
is conjugate to the generator ηω of the module
= ν(ω_ ω^ by virtue of the diagonal in the module M).
899
The proof of Lemma 12.6 for ET follows easily from the calculations L J for Ext . (,!•/, ΖΛ. For
the case (Er, dr), part (b) of Lemma 12.6 follows from the fact that the elements x, hl+2 must be zero
in Ext υ (U* (U* (MSU), Λ 2 ) on the b a s i s of § 7 .
2
Corollary 12.5. For MSU, tlie Adams spectral sequences^ (in U-cobordism and H ( , Z2)-theory)
are determined by the algebraic spectral sequences Er and Er.
In analogous fashion it can be shown that all known Adams differentials for A = P^ in both homology theories (the case of the homotopy groups of spheres) are also determined by Εr, Er and dr, dr.
By analogy with the case ρ > 2, bounds can be determined here also for the filtrations of elements
1
Ext v (see Corollary 12.3).
Appendix 1
On the formal group of "geometric" cobordism
(Theorem of A. S. Miscenko)
We consider an arbitrary complex X, the group U (X) and its subgroup Map (A7, MUx) C U2(X). In
what follows we shall denote Map(.¥, ll/i/,) by V (X). Since MUl = CP is an //-space, V (X) becomes
a group, which is communicative, and with respect to this law of multiplication we obviously have:
V(X)fa W(X,Z).
How is this multiplication in V (X) connected with operations in U (X) D V (X)?
As was already indicated in §5, we have
Lemma 1. a) If u, ν €• V (X) and (Bis the product in V (X), then the law of multiplication
= f(u, v) has the form
U<S>V=U-\-V-\-
where xt- € A ' l ( t + ' ' l ) = Sl\fl + ''i'> are coefficients
2
u@v
x
i,juivK
independent
of u, v,
b) u φ ν = ν 0 u,
c)(ii9«)9» = »®(p®i«),
d) there exists an inverse element u, where ΰ © u = 0.
The proof of this lemma follows in an obvious way from the fact that V(X) = fI2(X, Z) and the
possibility of computing all the coefficients on the universal example λ = CP . We note that xy l
= ICP1].
Thus, we have a commutative formal group with graded ring of coefficients Λ, and dim u = dim
ν = 2. As is known, the structure of such a group is completely determined by a change of variables
g over the ring Λ <£)? Q, "• — g(u) = Z_\l/iUl+i,
y0 = 1, such that
iSsO
g(u®v)=g(u)
900
S. P. NOVIKOV
Theorem (A. S. M i s c e n k o ) . The change
u -> g(u),
of variables
where
n+i
g ( u ) ^ 2J
u ,
iL
hjsO
ι J.
'
xn = L CPn] £ Λ" 2 ", reduces the formal group V ® ι Q to linear form g (u © v) = g (u) + g (v). Hence,
the change u -> g (u) reduces to linear form the formal group V (X)® Q for all X and uniquely determines the structure of the one-dimensional formal group V over the ring Λ.
Proof. We consider the ring U* (CPW) = Λ [ [ul ] and the multiplication CP^ χ CP^ -> CP™,
sending the one-dimensional canonical ί/j-bundle ζ over CP into ζ^ ® ζ2, where £;ly ζ2 ate canonical
bundles over CP χ CP. This multiplication induces a diagonal Δ : U* (CPW) - U* (CPW) ΘΛ U* (CF™),
which gives the multiplication in V{CP ).
Let u = g(u) Σλ;ω', where A(u') = u ® 1 + 1 ® u'. Then g is the desired change of variables.
We compute the coefficients λ ; . Let S(&) € /4
(see §5).
We have the easy
Lemma 2. The operations S^·, form a system of multiplicative generators for the ring S®Q. If
Ok (x) = 0 for all k, χ € Λ, then χ = 0.
Proof. We order the partitions ω naturally (by length) and consider
S(k)S<o(u,i · • • un) = S(h) ^ j ui'
· · ·us
ω = (ki,...,
SWSa(Ui
us+lo • • • °unr
ks),
2 « Λ . ( » 1 ·'. . »n) + a0S(k,(O)(Ui . . . Un) ,
...Un)=
where a0 φ 0, ω; = (hx, . . . , k^ + k, ki+1, . . . , ks). Since by the induction hypothesis all S ^ can be
expressed by the S^ ), the same is true for 5 ( ω > JJ. Since all 5 ω can be expressed by the S^-), the
lemma is proved.
We note the following equation:
We set
,li
V
„ (ft)
U' = 2 j A i
w
,
U
.
fc
V
(ft)
= 2 j Μ<
·
U
, i
'
Obviously, SÂu' = AS(fc)u', since t\u' = u ® 1 + 1 <8) u'. Since
i
i
j
we have
/
f~r A
/
^C^ " ^ 1 /
i
3
*
ι/·
7 \ Λ
\
i^) /
/
901
It obviously follows that for α. φ 0, β Φ 0 we have:
Σ (<*<fcA< +(i-
k)Ki-k) μ? = 0, / = α + β > 2.
i
Since μ<° = 0 for all i > 2, μ<° = 1, At = 1 and σ * 4 ) + (t - A j A ^ = 0, /t > 1, we have
S (σ(ίι)λι + (i — &)λί_^) μ/ = 0
i
for all /' > 1, and since
S
^j μ;· Xs = 6j, we have
i
Σ(σ<*)λί +(i — k)Ki-k)μ^λ™ = σ'^λί +(i)
k)Xi-k = 0.
i
Hence,
*
σ<Λ)λί = — (i — k) Xi-h.
Further, since σ* [ C P n ] = - (re + 1) [CPn'k} (see §5, Lemma 5), it follows that \ l = xt.,/i,
Xj = iCP'i € Λ' 2 ; satisfies the condition σ*4 λ; = - (i — A)A;.j. for all i, k. By Lemma 2, λ^ = λ;,
and the theorem is proved.
Remark. For a quasicomplex manifold X, the group V(X) is isomorphic to H2n.2{X) and the
meaning of the sum u φ ν is such that the homology class v(u) ν (ν) is realized by the inclusion of
the submanifold V1 ®V2, where u €. Uln.2{X), ν € V2n-2 ate realized by the submanifolds Vl, V2 C X.
Then the series
u Θ υ = u +· υ + ... =
/(H, V)
must be considered in the intersection ring i/* (X).
Appendix 2
On analogues of the Adams operations in i/*-theory
Analogues of the Adams operations Ψ^, £ A 0 χΖ l(l/k)\
were defined in §5 in the following
way:
b) ky¥ku (x) = Χ Θ . . . 0 Λ; (A times), where χ £ V (X).
Thus, the series Ψ^ has the form:
{)
g
where f(u, v) is the law of addition in the formal group V(X) and
*+1
(*) = •Σ
on the basis of Appendix 1, g'i(g(x))
= x.
902
S. P. NOVIKOV
From the associativity of the law of multiplication in V (X) follows the equation:
ψ£(ψύ(*))=ψ"(*).
Hence, always Ψ*° ψ£, = ψ*' in AU ® ZQ, since for any ra - °« and υ = B, . , . u n we have
ψ£,Ψ^ (u) = Ψ*'(«) by virtue of properties (a) and (b).
Of the assertions in Lemma 5.8, only part (d) is nontrivial, and it asserts that Ψν7 * (y) = k'-y,
y € Λ - = Q'J.
Theorem 1.* // a £ A is an arbitrary cohomology operation of dimension 2m, then we have the
following commutation law:
α
ψ£ = kmWu "a.
Proof. Let am = S(m) = Ε AU and u € V(CP™) C ί/2(ίΡω). Then
am(u)
1
f
=
u@...®u
since ιιφ... ® a C V(CP ). Hence, for the operations O( m) = S( m ) the theorem is proved. From
this Theorem 1 follows for all operations $(ω), since by Lemma 2 of Appendix 1 the ring S ® zQ is
generated by the operations S(/t).
Now let a € A'2m = U'2m (P). We assume by induction that for all operations in Λ" 2 ', / < m, the
theorem is proved. This means that for b € A'2', j < m, we have:
In view of the fact that Ψ*'*(έ,ό2) = Ψ*' * (&,)Ψ*' * (b2), the theorem is also proved for all decomposable elements of A'2m. Let a € A'2m be an indecomposable element. We consider Ψ^'*ajj (a)
= ^m-dim (^σω(α) by induction, for ω ^ (0). Since
ft
ft
we have
Ψυ'οΙ ( α ) = σ * (kma).
Hence, Ψ*'* (α) = hma, since
Π Kera^ = 0.
Since Theorem 1 is proved for Λ and S, it is also proved for A
= (AS) .
Thus, all assertions of Lemma 5.8 are proved.
We now consider an arbitrary ring K, the group of units U^ C Κ and A ® zK- We define the following semigroups in A
*From Theorem 1 it follows easily that all operations Ψ^, are well-defined over the integers on U° (X), as
in /(-theory.
METHODS OF ALGEBRAIC TOPOLOGY FROM COBORD1SM THEORY
903
1. The semigroup of multiplicative operations a £ A ® χΚ, where Δα = a ® a £ A ® Λ A ® z/£.
2. The semigroup of multiplicative operations of dimension 0,
AKCZAKCIAU®ZK.
3. The center Ζ χ C A\ of the semigroup Αχ.
4. The "Adams operations" Ψ£, £ A\, where q € U ^ (the group of units), defined by the requirements of Theorem 1:
ψ£ (α) = q-maW£,
dim a = 2m,
J u s t a s earlier, a multiplicative operation a £ Αχ i s defined by a series a(u), u € U2(CP
) ®ZK
the canonical element, α (it) £ Α χ L L iz-J J , Λ ^ = Λ ® χΚ.
We now consider the question of defining the Adams operation. L e t Κ = Q LfJ, Λ ^ = Λ ® z/\. We
consider for a l l integral values I the series Ι Ψ^ (u) £ U* (CP°°), defining the series «Ψ^, (u) e LîCP™
®ZK.
Remark. If Κ is an algebra over Q, then the Adams operations Ψ^ £ A ® %K are always defined,
since the series «Ψ8 is divisible by £ and Ψ[, ( u ) ς (/* (CP) ® ZK.
Theorem 2. a) for any algebra Κ over Q without zero divisors and for Κ = Qp, Z, the "Adams
operations" Ψί £ A ® z ^ are defined, where a € Κ in the Q-algebra case and a €. Up in the case
Κ = Qp Ji.e., Up = UQ S, a = ± 1 in the case Κ = Z, such that:
11; <υαιψα2 - φ α ι α 2
υ υ ~υ
2) Ψ^'* :Λ-Κ2ί - Λ"κ2ί is multiplication
by a ' .
3) Ψ^ ° a = α-'αΨ^, wAere ο 6 . 4 ^ ® Z X is of dimension 2i.
4) 7'Ae series αψ?' (u) for u €. V (CP ) makes the operation of raising to the power a, a C X*,
well-defined in the formal group V.
b) 7"Ae collection of all Adams operations forms a semigroup Κ = Ψ(&) for a Q-algebra K,
Ψ (Κ) ~ Up for Κ = Qp, Ψ (Ζ) = Z 2 , which coincides precisely with the center Ζ κ of the semigroup A0^
of multiplicative operations of dimension 0 in the ring Ay® Κ for Κ = Qp, Z, while for a Q-algebra Κ
the center consists of^(K)
and the operator Φ, where Φ (u) = g(u).
Remark. Although a C A% is such that Δα = a® a and is given by a formal series beginning with
1, where a(u) = u + . • • , still the coefficients of the series lie in Λ or Λ®Κ, while the law of superposition of series ax · a2(u) takes into account the representation of AU ® Κ on Λ ®K. Hence AK is
not a group (as usual in formal series of this kind), but a semigroup. An example of a "noninvertible"
element α £ .4^ is given by the series
where Φ2 = Φ and Φ* (y) = 0, γ £ Λ 2 ' for / > 0.
We prove Theorem 2. Part (a) was essentially already proved above. In order to establish that
= Z ^ , we consider an arbitrary element α £ Z^ and we shall show that α £ Ψ (Κ). Since the
904
S. P. NOVIKOV
series a (u) = u + . . . , we have α*|Λ° = 1 and a*\A2 is multiplication by a number α Ε Κ. If a*\A'2> = 0
for all / > 0, then it follows that a = Φ* and hence α = Φ, while Φ ξί Α ® zQp·
It will be assumed
that for some /, α |Λ" 2 ; Φ 0, / > 0. If α \A'2' is the operator of multiplication by a number &·, then it
is easy to see that A; = k[ and α = Ψ^1, where kl £ K* or kx € i/ p . We shall show that for all / the
operator a is multiplication by a number kj. If /„ is the first number for which α |Λ' 2 '° is not multiplication by a number, then, nevertheless, on the decomposable elements A'2J° C Λ" 2 ' 0 , α is multiplication
by a number in view of the fact that a (xy) = a (x)a (y). If y 6 Λ" 2 ' 0 is an indecomposable element,
then a (y) = Ay + y, γ C Λ and γ = 0. Let b £ A°K be such that b (γ) = μγ + γ, where y €. Λ, γ φ. 0.
Then b α φ. a b on Λ" 2 ; °, which is impossible. The theorem is proved.
Appendix 3
Cell complexes in extraordinary cohomology theory. f/-cobordism and A-theory
Let X be a homology theory with a multiplicative stable spectrum, X ® X -> X, and let Λ* = X (P)
be the cohomology ring of a point. We require that Λ be a ring with identity. We note that Λ = X*(P)
is also a ring, and we have the formulas Λ = Hom*^t (Λ*, Λ*) and Λ* = Hom^ (Λ, Λ). Obviously, the
rings Λ and Λ* are isomorphic and Λ 1 = Λ*" £ , Λ 1 = 0, i < 0.
Let Κ be a cell complex and KL C Κ be its skeleton of dimension i. We construct a "cell complex
of Λ-modules"
SX(K):
a) if dim Κ = 0, then Sx (K) is a free complex Σ Λ (P.), where the Ρ are the vertices of Κ and A(P)
is a one-dimensional free module with generator up : we set dup = 0.
b) Suppose that for all K1, j < i, Sx(K')has
been constructed so that βλ = λθ, λ €. Λ, and the genera-
J
tors of Sx (K ) ate in one-one correspondence with the cells of K1.
We consider the pair (K1, K1'1),
where K'/K1'1
is a bouquet of spheres S i \ / . . . \/Sl
. We adjoin
1 1
to Sx (K ' ) free generators « , , . . . , uq of dimension i. A differential in the complex Sx (Κ1'1) + Λ ^ )
+ . . . + A(uq ) is introduced as follows:
i) dk = xd, λ e Λ;
2) (9u; = Zj β S^ (K''1),
1
where z ; is such that dzy = 0 in S^ (K1'1) and the homology class L z ; J
1
6 A"* (X * ) is represented by the element equal to <9u;, where d : X* (S[ \ / . . . ν Slq .) -> A"* (K1'1) is
the boundary homomorphism of the pair (X1, X1"1) and Uj ε Χ*{Κι/Κ1'1)
corresponds to the sphere Slj.
Thus, a complex SX(K) of free modules arises.
Lemma 1. The complex Sx (K) is uniquely defined up to the choice of the system of generators,
and the differential dinSx
coincides up to higher filtration with the homology one.
Obviously,
H(SX (K), d) = X* (K) as A-modules.
For a cellular map Yt -« Υ2, there is defined analogously a morphism of free complexes Sx (Y2)
-» Sx (Y2), also unique.
Let Υ = Υ\ χ Υ2 with the natural cellular subdivision.
Question. When is there defined a pairing
Sx(Yl)ÂSx(Y2)^Sx(Yi
which is an isomorphism of complexes?
X Y2),
905
Now l e t X = U.
Conjecture.
For a pair Yl, Y2, the complex Sy (Y1 χ Υ2) is homotopically equivalent to the
tensor product
Su{Yi)®iiuSu{Y2).
Let A be an arbitrary fl(/-module. The homology of the complex Sy&o Â we shall denote by
U* (Y, A), and the homology of the complex HoniQy (Sy, A) by U (Υ, A) (cohomology with coefficients in A).
We shall indicate important examples:
1. A = Q,y is a one-dimensional free module.
2. A = Ζ ιχ\, dimac = 2, and the action of ily on A is such thaty(% m ) = T(y)xm+i,
where
y €. Ω;/, 2£ = dimy, and Τ (γ) is the Todd genus. Here A i s a ring and there is defined a homomorphism λ : Ων -> Ζ [χ], such that λ (y) = Τ (γ)χΚ
3. A = Ζ [%, χ-ι], where d i m * = 2, dim χ" 1 = - 2 and xx'1 = 1. Here 4 is a ring, while Ω{7 a c t s
on A just a s in example 2: y ( x m ) = T(y)xmJri,
- °o < m < oo.
4. /I = Z, where Ζ = Ω(//Ω{/, Ω(/ is the kernel of the augmentation ily -> Ζ and the action of
Ω;; on Ζ is the natural one.
C o n j e c t u r e . For the Q,y-modules
U ( , A) are isomorphic,
groups
k*-theory
for A = Ζ [χ], to unstable
A = Z. The homology
theories
A = Ω(/, Ζ LxJ, Ζ Lac, x~'J, Z , iAe corresponding
respectively,
K-theory
to the cobordism
theory
cohomology
U for A = ily, to
K* for A = Ζ [χ, χ·1] and to the theory
stable
H* ( , Z ) for
Zlx, x~l\, Z, are isomorphic,
U*( , A) for A = Ωΐ), Zlxi,
respec-
to {/*, A*, K* and //* ( , Z ) .
tively,
Theorem 1. Since the complex
spectral
with term E2 - Ext^*,, (i/* (Y), A) which converges
sequence
a spectral
sequence
with term E2 - T o r * ^ (U* (Y), A) which converges
Theorem 2. Since the complexes
are complexes
dimensional,
(Sy (Y), d) is a complex of free Q,y-modules,
of free Α-modules
Sy (Y)(g)
.,Zlx\
Ω
there exists
to U (Υ, Ί ) , and there
universal
coefficient
exists
to ί/*(Υ, A).
= S/C(Y) and Sy (Y) (gjjj
Zlx, ar 1 J -. S/C(Y)
for A = Ζ lx\, Zlx, %"'J, and the ring Ζ [χ] is homologically
we have the following
a
one-
formulas:
1 ) 0 - » - Extzi* ] (k., Ζ[*])-*• k* -»- Homzcr] (*., Ζ[«])-> 0,
2)
0-»-*.
®z[ I ]Z[a: > x- 1 ]->^.-»-Torz [ * r ](A;. 4 Z[a;,ar- 1 ])->0,
3) 0-*-Λ
where in formula 1) A* awe? k are connected,
module, and in formula (3) A* and //#, since
in formula 2) k* and /£*, since
Ζ is a Z
Ζ Ix, %"'] is α Ζ \_x\-
Ixj-module.
It i s p o s s i b l e t o find a number of other formulas c o n n e c t i n g A*, A , Κ*, Κ , //*, // a n d a l s o
Kunneth formulas for the d i r e c t product Υ, χ Y2, s t a r t i n g with t h e complex Sy ® n Ζ L%J a s a
Zlxl-
module a n d the fact t h a t Ζ [χ] i s o n e - d i m e n s i o n a l , a s i s Ζ.
We note a l s o that H o m ^ ^ Z Lac], Ζ [ χ ] ) = Ζ [ y ] , where d i m y = - 2.
The author has available a derivation of Theorems 1 and 2 from the Adams spectral sequence in cobordism theory, and hence Theorems 1 and 2 do not depend on the preceding conjectures.
906
S. P. NOVIKOV
In all the formulas of Corollary 2 one can start from the complex Hornby (Srj, Ζ Lxl), which is
a complex of free Ζ LyJ-modules for k -theory.
With the help of the complex Srj (Y) it is possible to introduce, in addition to the cohomological
multiplication, also the "Cech operation" f) such that (o f] b, c) = (a, be), where c, a E U*, b € {/*
and a f| b Ε £/*, while (a f] b, c) Ε VLJJ . Analogously for k*- and k*-, K*- and K*-theories.
The Poincare-Atiyah duality law, of course, is treated in the usual way by means of the fundamental cycle and the Cech operation.
We note that the homomorphisms at, introduced by the author on the ring Ω[/ represent "characteristic numbers" with values in Ω(/, since the scalar product lies in Ώ,υ.
Appendix 4
i/*- and A*-theory for BG, where G = Zm.
Fixed points of transformations
In this appendix we shall consider the following questions:
1. What are the cell complexes Sv (BG) and Sk (BG) for G = Zm?
and k* (BG), where A = Ωυ and Λ = Ζ [χ] ?
What are the Λ-modules £/* (BG)
2. How to compute in U-^(BG) the following elements: let the group Zn act on Cn linearly, and
without fixed points on Ca\0, i.e., by means of diagonal matrices (a;/), where a I ; = exp(2nixj/m)
and Xj is a unit in the ring Zm. Then an action of Zm on S 2 "" 1 is defined, and by the same token a
map f%i, . . ., xn : S2n'i/Zm
-» BG, which represents an element of U2n.1(BG). This element we denote by an(xl, . . . , xn) €. U2n.l(BG). It is trivial to find a n ( l , . . . , 1) ("geometric bordism") and to
show that an (xu . . . , χη) φ 0 for all (invertible) xx, . . . , χη Ε Zm (see [ ]),
ν α η {χι,
...,χη)Φ
0,
ν:
Ut
->• / / , ( , Ζ).
This question arises in connection with the Conner-Floyd approach to the study of fixed points
(see [ " ] ) .
We consider the question of computing the cell complexes SJJ (BG), S^ (BG) and S^ (BG) (see
Appendix 3).
We recall the well-known result of Atiyah [ 7 ] that Kl (BG) = 0 and K° (BG) = R(jGA, where
RJJ (G) is the ring of unitary representations. For G = Zm, the basic unitary representations p0 = 1,
pi = \l27Tl m\, . . . , pk = \l27Tl m I, . . . , pm ί are one-dimensional, while as a ring a generator is
pi = ρ with the relation pm = 1. By virtue of this we can choose in K" (BG) an element t, corresponding to ρ - 1, with the relation Ψ"1 (t) = 0, where Ψ™ is the Adams operator.
We consider the ring k* (P) •= H o m * [ x ] ( Z [x], ZVx\) = Ζ [γ], dimy = - 2. We have
Lemma 1. The Z. [y\-module k (BG) for G - Zm is described as follows:
a) k2i+l
= 0.
2
b) k ' (BG) is isomorphic to the subgroup of k° (BG) consisting of elements of filtration > 2/, and
this isomorphism is established by the Bott operator y':
c) The action of the rings Βψ and Αψ is well defined on k (BG).
907
2
d) There exists a natural generator u £ k (BG) such that every element of It' (BG) has the form
m
^ y"} U<lj and there is the relation (m^ )(u)
1
= 0, or Ψ" (yu) = 0, ivhere yu € k° = K° is the canonical
j
generator t C K° (BG); and we have the equation
The proof of the lemma follows easily from the results mentioned about K° (BG) and the discussion of the spectral sequence with term E2 = Η (BG, Ζ lyi) which converges to k (BG).
/ τη \
1
We denote the expression πιΨ" (u) by Fm(u) = 2
(
M ~ y)* ' " * · From Lemma 1 follows
Lemma 2. The cell complex S^ (BG) = Hom^^^S^, Zlxi) of modules over Ziyi is a ring with
multiplicative generators (over Ζ lyi) ν (of dimension (\))andu (of dimension (2)) and additive basis
\ysun, yqvu1}. The differential d in this complex satisfies the Leibnitz formula, commutes with multiplication by y and has the form:
du = 0>
The cell
complex
of k-theory
Sic (BG) = H o m ^ r y](Sk>
of free
S/C(BG)
Ζ Lyi), while
for G = Zm
Sjc(BG)
over
dv=Fm(u).
in the natural
Zlxi,
cellular
subdivision
= H o m ^ r -,(Z lyi,
Zlxi
has the
Zlxi),
form
is a
complex
modules.
Lemma 2 follows easily from Lemma 1 and Appendix.
We turn now to i/*- and U -theories. For the element u €. V (X) = Map(/Y, Μϋγ) C U2(X), the
series m!Vm,(u) = g-*(mg(u)) (see Appendix 2), where g(u) = 2
U
iCPn]un+l/(n
series" (see Appendix 1). We denote the series ίπΨ^ (it) by Fmjj(u).
+ 1) is the "Miscenko
Let
Λ = U*(P)= ΗοτααΌ(Ωυ,Ωυ),
21
2
and let S\j (BG) be the cell complex in U -theory which is a complex of Λ-modules, with Λ' = Ω '.
With the help of the Conner-Floyd homomorphism σ^ : k° -•> U2, k° - K°, we obtain from Lemma 2
Theorem 1. The cell complex (in the natural cellular
subdivision)
SU(BG)=Homau(Su,Qu),
which is a complex of free Α-modules, Λ = U (P), with differential d, is a ring with multiplicative
generators ν (of dimension (1)) and u (of dimension (2)) over Λ, given in the following way:
<;2=0,
d(v)= Fm,u{u),
d(u) — 0.
The complex Sy (BG) is isomorphic to Homt\(Sy, Λ), C = Zm, where ily = Hom*v(A, Λ). The complexes Srj §§Q Ζ LxJ and S ®Ω Zlx, x~ii are isomorphic, respectively, to the complexes Sj (BG) and
S/C(BG) in k- and K-theories.
We pass now to the automorphisms of the complex BG -> BG. Such automorphisms for G = Z m are
completely determined by automorphisms of the group Zm — Zm, which are multiplication by k, where
4 is a unit in Z_ .
908
S. P. NOVIKOV
There arise automorphisms
Xh: BG-+BG,
where λ^ is completely determined by the images
since λ^ is a ring homomorphism which commutes with the action of Λ.
We have
Theorem 2.
The homomorphism of multiplication
by k
for G = Zm and (k, m) = 1 is a ring homomorphism which commutes with d and multiplication
by Λ and
is defined by the following formulas:
a) ύ ( « ) = Fk,u{u),
The proof of Theorem 2 is obtained from the fact that under λ/,, ("geometric cobordism") u €. U2(BG)
must go into ΑΨ (u) by definition of the operator Ψ . This implies part (a). Part (b) follows from the
fact that (A.J. (υ) = λ/j dv = D (u) v, where D (u) is a series of dimension 0 with coefficients in Λ.
We now pass to the question of fixed points of transformations Zm. Let Zm C Zm be the multiplicative group of units, xlt . . . , xn € Zm and an (xi, . . . , xn) € U2n.l (BG) the element defined by the
linear action of the group Zm on 5 2 "" 1 C C"\0 by means of multiplication of the /-th coordinate by
exp(2nixj/m),
Xj €. Zm. There arises a function
*
*
an' Zm X . . . X Zm —y U2V.—1 (BG).
Let m = pn, ρ a prime and mt = p " ' 1 . Then Zm
U*(BZni)-^
U*(BZm).
C Zm and there is defined a homomorphism
We have
Lemma 3- Given a quasicomplex transformation Τ : HI" —> Mn of order m which has only isolated
fixed points Pl; . . . , Ρq, we have the equation
1
2
α η ( χ υ · , . . . , xnj) = 0 mod U, (BZmi),
where the %i;- are the orders of the linear representation
of the group Zm at the point Ρj {clearly,
Xij € Z * ).
This lemma for prime m = ρ was found by Conner-Floyd L J (here, mt = 0), and it is trivial to
go over to m = ph.
*A11 homological deductions from Theorems 1 and 2 of this appendix can be justified, without the complexes
j, merely from Theorems 1 and 2 of Appendix 3·
909
It i s e a s y t o s h o w t h a t for a n y (xi, . . . , xn) €. Zm χ . . . χ Z m
mod U.
un (xu ...,χη)φ0
(BZm<),
whence follows the theorem of Conner-Floyd-Atiyah: there does not exist a transformation Τ with one
fixed point. For ρ > 2 this is also true for real transformations T, as can be seen from the analogous
application of the theory SO* & Ζ [ 1 / 2 ] .
We now pass to the question of calculating the function a-n (xi? . . . , xn) £ U2n-i ( ^ Z m ) . We denote
υ
by 2π-ΐ ^- ^ 2 η · ι ( ^ )
t n e
so-called "geometric bordism" a n ( l , . . . , 1). In the complex Sy(BG)
scribed in Theorem 1, the element i^2,1.1 is adjoint to vu""
1
de-
1
€ Sy(BG),
i.e., (i>2n.,, iû™" ) = 1, (;.';„.„
1111»-' + ' ) = 0 for A; > 0, where χ 6 Λ*.
We shall calculate the function a n (x,, . . . , xn) by the following scheme:
1) Clearly, a , (x) = xt,, C f / 1 ( S G ) - Z m .
tn
2) If
I
x
Zj «ftiîji · · · 1 hi) ^
3=1
0,
ZJ an(Xij,
3', s
2j an-k(i/ft4-l,j. · . · , i/n,j) ^
j=i
0, then we have the equation:
• • • , Xhi-, yh+i,s, · · · , JJn.s) =s 0 mod Z?Zm,.
This follows in an obvious way from the fact that transformations 7\ : Μ -> 1/
and 7'2 : ,1/ -> ,W induce
π
(7\, Γ 2 ): Λ/ χ Λ/ " , where fixed points (and their orders) correspond to each other.
3) If λ χ : BG -> BG is induced by multiplication by χ £ Z m , then an(x,
. . . , x) = kx>t
(v2n.l),
where
the structure of λ χ „ is described in Theorem 2.
As examples of the application of this scheme we shall indicate the following simple results:
Lemma 4. If v. 6'* -> //* ( , Z) is the natural homomorphism, then we have the equation
V(/ n (Zl» . . . , Xn) = (Xl • • • Xn) V (ϋ2η-ΐ) ,
where v(vln.l)
€ fl2n-i(R^m)
= Zm is the basis element.
L e m m a 5 . For η = 1, 2, 3 ">e Aaue iAe
formulas:
" ^
[CPl] vs.
From Lemma 2, in an obvious way, follows the corollary on the impossibility of one fixed point.
Now let m = p, where ρ > 2 for η = 2 and ρ > 3 for η = 3· Under these conditions, by the scheme
indicated above, one easily obtains from Lemmas 2 and 3
Theorem 3- The functions
an (at,, * 2 , . . . , %„) /Or π < 3 /*as iAe following form:
fli(x) = XVi
u
(obviously);
x2) = {XiX2)v3;
ft3 (#1, Z2, ^3) = (a-'l^2^3) Vs + '
·
[CP1] VS.
910
S. P. NOVIKOV
n
Suppose given a group of quasicomplex transformations Z p : M -> M" with isolated fixed points
Pu . . . , Pq at which the generator Τ 6 Zp has orders % l ; , . . . , xnj G Z p , j = 1, . . . , q, where x^j € Ζ
x
We consider the point (ΧΙΛ, • • • , Xkj, • • • > nq) G Zp" up to a factor μ C Z p , μ φ 0. Thus, {χίΛ,
qn
l
G P ~m
• • •
qn
xhe group Sn χ Sq, where S^ is the group of permutations of k elements, acts on P ~ .
n
Definition. By the type of the action of the group Z p on M with isolated fixed points we shall
mean the set of orders of (xli, • • • , Xkj, • • • , Xnq), of any generator Τ G Z p , considered in the proqn
1
jective space P ~ j
factored by the actions of the group Sn of permutations of orders of each
point and the group Sq of permutations of points.
From Theorem 3 follows the
of Ζp on Mn is given in
Corollary. For p> η and for η = 2 , 3 , the set of types of actions
by the set of
Pqa~l
equations:
Σ
π, IT. •
τ
Λ— 0
-r- =iL· Π
h
9 "ί
where the xsj are the orders at the point Pj and oîXij, • • • , xnj)
i s
η
the elementary symmetric poly-
nomial.
Appendix 5
The conjecture on the bigradation of algebraic functors
in S-topology for all primes ρ > 2
In the Introduction and also in §12 the possibility was already discussed of the appearance of a
new categorical invariant — an additional grading, connected with the Cartan type, in the Adams spectral sequence for ordinary cohomology modp, ρ > 2, from which it would follow (see the Introduction)
that the homotopy groups in the category of torsion-free complexes could be formally computed algebraically by the theory of unitary cobordism. We shall formulate here more exactly the corresponding
conjecture.
First of all, we shall go to the question of the category S®zQP
for ρ > 2. Let Κ (π) G S be the
spectrum Κ (π, η). The following fact is known (H. Cartan): the Steenrod algebra A = H* (K(Zp), Z p )
is bigraded:
1 = Σ/4 ' , where dim = k + β and β is the type.
Conjecture: I) Let the bigradings H{X, Z p ) = E// A 'ând Η (Υ', Ζρ) = I / Z ^ b e well defined,
and let the morphism f: X -> Υ in the 5-category preserve the bigrading. Then in the exact sequences
and
for the objects Z, Z' £ S, the bigradings of the functors H**(Z, Zp) and //** (Z', Z p ) are well
defined, and the exact sequence of the triple (Χ, Υ, Ζ) is
6
/·
,... ->- H > Ρ (Χ)-> Η"· Ρ+ (Z) -+ H - P+ (Y) -+ Hh· P+1 (X) ->-...
h
1
h
1
II) For X = K(Zp) the bigrading coincides with that of Cartan.
911
III) The cohomology A -module // (A, Z p ) is Jjigraded, if in // (Α, Ζp) the bigrading is well
defined.
IV) All these properties are fulfilled in the subcategory Sgr C S<S>zQp obtained from K(Zp)
inductively by means of bigraded morphisms and passage to "kernels" and "cokernels"
here Qp is the p-adic integers.
Assertion 1. If the conjecture is true, then the spectra of points (spheres) Ρ and complexes
without p-torsion in homology belong to the category Sgr.
2. If the analogous conjecture of bigradation for other functors (for example, homotopy groups)
is true, then the entire classical Adams spectral sequence and the stable homotopy groups of
spheres for ρ > 2 can be completely calculated by means of the theory of unitary cobordism by the
scheme described in the Introduction and in §12. In particular, we should have the equation:
π?.' (S») « Ext" (Λ, Λ) ®zQp, Ρ > 2.
A
This means two things: a) the triviality of the Adams spectral sequence constructed by the author
in the theory of i7-cobordism; b) the absence of extensions in the term Eco = E2.
3· For ρ = 2, the conjecture in such a simple form is trivially false. {In the spectral sequence
for the stable groups of spheres, all powers r\k ^ 0 for an element η representing the Hopf map in
7Γ, (S), hence t]h for h > 4 must be killed off by differentials.\
4. The classical Adams spectral sequence with second term E2'
for X Ε Sgr is arranged as follows:
d • Fs'h- β_>_ Fs+r'
h
'
pifr
= Ext4' '^(//* (A), Z p )
~1
We note that h0 Ε E x t ^ 0 1 ( Z p , Z p ) is associated with multiplication by ρ (ordinarily we have h0
Ε Ext'/). Here, the dimension differs slightly from that described in §12 by a simple linear substitution.
Examples of bigradation (the simplest). Let A = K(Z) + EK(Z) and Υ = Κ(Ζ q). From the ordinary point of view we have:
H*(X,ZP) =H'(Y,ZP) =Α/Αβ(η)
+Α/Αβ(ν),
where dim u = 0 and dim υ = 1. However, for X the ordinary Adams spectral sequence is zero, but
for Υ we have: dq (v ) = hû , where u C E x t ° / and ν € E x t / , since 77* (Υ) = Ζ q.
From our point of view the situation is thus:
01
0
0 1
a) H**(X, Z p ) Α/Αβ(α) + Α/Αβ(ν), where u €. //°'°, ν € Ζ/ ' . Hence u* C E x t / ' , ν* Ε Ext '
and hoU G E x t 9 ' 9 ' 0 ; by dimensional considerations, dq (ν ) Ε E x t 9 ' 1 ' 9 " 1 ,, and
E x t 9 ' 1 ' 9 ' 1 = 0.
andExt
b) #**(Y, Z p ) = A/A/3(u) + Α/Αβ(ν),
Φ 0 for i = q.
u 6 H°'°, ν € Η 0 · 1 , then υ* € Ext 0 /' 0 and v* € E x t ° / ' \
Besides the facts indicated earlier, there are subtler circumstances which corroborate the
conjecture:
912
S. P. NOVIKOV
1. From the results of the author's series of papers on the /-homomorphism / * C 77* (S ) and
the results of the present paper, it follows that Ext''^(A, h)®zQp
consists (for ρ > 2) of cycles
for all differentials, while elements of Ext 1 '^ are realized by elements of πίρ)
order; moreover, nlp)(SN)
= E x t ^ + . . . , where Ext 1 '* = / * ®
(SN) of the same
QP.
Z
2. The Adams spectral sequence in U- theory would not have to be trivial from dimensional
considerations (obviously, only dt is zero for i - 1 Ξ 0 mod 2p - 2). There first appears an element
χ € E x t 2 ' ^ 2 ^ " 0 where d^.^x)
= ?, since Ext 2 p J' 1 ' 2 p 2 ( p ' l )
+ 2p 2
" ^ Q_
In realitV; these
elemerits
in
(/-theory are "inherited" from ordinary cohomology theory Η ( , Z p ) together with the question
about d2p.1 (%). A few years ago L. N. Ivanovskii informed the author that with the help of partial
operations of Adams type he had succeeded in showing that d2p.l(x)
= 0 for ρ > 3 (?). However,
neither Ivanovskii nor the author were able t r . verify this calculation, and hence this fact remained
obscure. Recently Peterson informed the author that it has only recently been proved by the young
American topologist Cohen L. J for all ρ >_ 3 (more exactly, the analogue of this question in the
classical theory, from which, of course, it follows).
3. The fact that the "algebraic" Adams spectral sequence associated with the "topological"
one, which begins with E2 = ExtJ**
( Z p , Zp) and converges to Ext**^ (Λ, Λ) <g) zQp
(see §12), is
algebraically well-defined, is non-trivial a priori. The situation here is that if for some spectral
sequence (Er, dr) we consider the complementary filtration in E2 and define on all the Er associated
differentials dr, then very often the dr are not included in a well-defined spectral sequence (of
algebras). Hence the fact of such a well-defined inclusion is in our case an extra geometric argument for the existence of an invariant second grading in the subcategory Sgr C S ® zQpReceived 10 APR 67
BIBLIOGRAPHY
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the methods of algebraic topology from the viewpoint of cobordism

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