THE METHODS OF ALGEBRAIC TOPOLOGY FROM THE...

THE METHODS OF ALGEBRAIC TOPOLOGY FROM THE VIEWPOINT OF COBORDISM

THEORY

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Izv. Akad. Nauk SSSR Math. USSR - Izvestija

Ser. Mat. Tom 31 (1967), No. 4 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 sequencein cobordism theory, calculation of the ring of cohomology operations in this theory, andalso a number of applications: to the problem of computing homotopy groups and the clas-sical 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 Eilenberg-

Steenrod 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 ac-

count 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 com-

plete 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 in-

variants 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, Conner-

Floyd, 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 argu-

ments 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 cobor-

dism 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 homolog-

ical 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], Λρ = 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 Λρ, lies in A V <g) z Qp D A V, and Λ ® ζ Qp is a local ring

with maximal ideal m C Λ ® z Qp, where h ® z Qp/m = Ζp. Note that Λρ is an A ^-module and A u is

also a left Ap-module.

Consider the following rings:mv = m Π Λρ, 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 thatH* (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 829

A = AP

U = 2 τηρ*Αρ

υ/τηρ**Αρ

υ,

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 Λρ and the induced filtrations.

It turns out that for all ρ > 2 the following holds:

Theorem. The ring Ext-J" (Λρ, Λρ) 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*{ , Zp)-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> zQp-

We 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 * * *

= Ext^u (ΛΒ, Λρ) = ExtA (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 Ext^iZp, 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 alge-

braic 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 (E r, dr), if the sequence (Er, dr) is

trigraded by means of the Cartan grading, as is {Er, ar). Therefore the orders \n (SN)\ would coin-

cide 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 connec-

tions 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 ques-

tions 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;A —>B Β —>G

b) for commutative diagrams 1 [ or \ 1 there exists a unique map of short exact sequencesA'-*B' B'->C'

(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).


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 se-

quence 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 defini-

tion 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 mor-

phisms 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 Η

Ζ Τ

ζ + 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 (direct)

fn· • ·fn+h-l'· Xn+k-*~EhXn (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 imHom*(X n , Υm)

832 S. P. NOVIKOV

in the case of direct spectra and

Horn* (X, Y)= lim lim Horn* (Xn, Ym)

m η

in the case of inverse spectra. Here, of course, let us keep in mind that in taking limits the grading

in Horn* ( , ) is taken in the natural way. As usual, remember that the dimension of a morphism

Ε 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 inclu-

sions S -> S and S -* S. We have the simple

Lemma 1.1. In the categories S and S there exist short exact sequences 0 - * / l - > B - > C ^ 0 , where

A, B, C GS or A, B, C CS, satisfying axiom 1 of the category 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 S

and S there exist direct sums with amalgamation satisfying axiom 3.

Proof. The existence of direct sums with amalgamation in the categories S and S is proved im-

mediately.

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^ -> C ' ^ - ^ 0) .

By axiom 1 of the category S we have spectra in 5, C = (Cn.) and C = (C' m ^) and morphisms

C -> A and Β -> C. 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, for the homology theory by

Αχ. They are graded topological rings with unity. ^

Note that an infinite direct sum Ζ = X Z ; of objects Xi CS l ies, by definition, in S, if we let

Zn = Σ Xi and Zn -> Zn.i be the projection. Obviously, Χ* (ΣΑ^) is an infinite-dimensional free

Ax-module, being the limit of the direct spectrum

Horn' (Zn, X) - • Horn* ( Z n + 1 , X),


< -

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 -^' -*" 2 ^ i -*-··· J

«~

where A ; is ΕΎίΧ, and therefore l ies in S, and the ,4x-module Horn* (Χ, Έ.Χ;) is free.

By X-free objects for X € S we mean direct sums Σ^f£, where X; = £ y ' A for arbitrary integers

Ji- Finite direct sums belong to S.

There are simple properties which give the possibil i ty 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-i gX 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 (Cx)£ = X* (Z t) and (θΓ)ί = T*{Zt) and the differentials (9*: (CJ^CC*);. ,

and δγ:(Βγ)ί-*(Βγ) i + 1 are the compositions

a z : X* (Zi) ii- Χ* (Fi-i) - i X* (Zi-t)a n d 5

δΓ: 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

qa: Hom*(7\ Y) - E0^* , 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 in this spectral sequence coincides precisely

withExt**AX(X*(Y),X*(T)).

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) —>-X*(Y)i 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. There-

x*(foi *fore Χ* (ΣΧη .) -> Χ* (Υ) *-X* (Υ) is an epimorphism. Consider the short exact sequence

i

Obviously X* (Y0

(n^) = KetX* (f0) and Yo

( n > GS. Now take a large number n1 » η and do the same to

^o as was done to Y, and so on. We obtain a filtration

v _, v(n) 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-t

AX(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), Χ* (Τ)) arises

in the following cases:

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


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 E2

S' l = Ext^A5» '(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: Εr

s·' -» Εr

s + r > ' + r ' 1 and the groups

_2J En are connected to Homm (T, Y) in the following way: there exist homomorphismst—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^^, (X* (T)y X* (Y)). In

this spectral sequence, dT: Erp' q -» Er

p'r' q + r + 1 , 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(77, 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 (p-

adic integers). The respective categories will be denoted by So for G = Q and Sp for G = Qp, ρ a

prime.

3. In S (or Sp for ρ > 0) we single out the subcategory D (or Dp (_ Sp) 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\/ Υ if

Ζ = 0 (a point).

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

Z *-v XT- \r γ rrQy A. — ~ Λ., Λ. t 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). Since X is multiplica-

tive and Ρ ® Ρ = P, we have a multiplication

Η η Χ Hm-*~Hm+n-

Let 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-


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 ΙΗ°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 multiplica-n->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 + n

by means of the multiplication in the spectrum X.

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 particu-

lar, 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 men-

tion 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, σ2 = 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 equalto 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 ca-

nonically isomorphic to the homology Adams spectral sequence (E'r, d'r) with term E'z = Ext/icrA-

(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 + l for direct spectra. Analogously

for inverse.

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 fol-

lows that all Ext s

χ( , ) (g) zQ = 0 for s > 0, since Ext^A"' £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 cate-

gory 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 re-sults. We note also the peculiar analogues, first discovered and applied in L2^J, to the Steenrod powers in thecohomology 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 inmore recent literature on these questions, although they are of value; for example, they reflect on the multiplica-tive formulas of Theorem 2 in L^J for ρ > 2.


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 Eckmann-

Hilton 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 peri-

odicity.

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 = BO χ Ζ, and all k0n are (n-l)-connected. We have kO ^

= (.k0j-lh, where nSnkoJ-n

1.} = 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 coincide up to suspension,

and the spectrum It is a sum of two spectra of the type k = kO + E2kO .

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 investi-

gated. All of them are multiplicative spectra and the corresponding cohomology rings have commuta-

tive multiplication with identity. Let us mention the known facts:

2) MS0®z<?2 = S Ε

λίΚ(Ζ) + Σ Ε

μ«Κ(Ζ

2) [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 multiplica-

tive spectrum since MTm + n = MTm (g> MTn.

Let us mention the structure of the cohomology M*(p )(/>), where 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:X^EyX 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, sincek

MU <S> zQp is a s u m °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 multiplica-

tion 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 € Dp 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 com-

plexes 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).


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 -> Fo -> 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 m-

connected in Dp. If X = iW(p), then it may be shown furthermore that in constructing Co in correspon-

dence 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 ob-

viously 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)

i n § 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 k-

connected 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

(p),M(p), since

The theorem is proved.

Note that for X = MU, M(p), stable spectra Υ and finite complexes T, all groups Ext 5 ' ^ are tor-

sion groups for s > 0, as derived in §2.

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), state-

ment (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'')CEs

r

+ 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 = Q for

t < 2s(p—1) and for f ^ 0 mod2p — 2, and the differentials dr on the groups Er ® zQp 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

AU. In the next section this ring will be completely calculated.

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 as a Λ-module is asso-

ciated with U* (T), where Εχ = Εk. 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(Ep

r· q) C £? + r- «*r + 1 . Denote by Λ^

C Λ the subring of polynomials in generators of dimension < 2/V, Λ = U* (P) = Qy. The ring Λγ is

Noetherian. Similarly, let Λ C Λ be the subring of polynomials in generators of dimension > 2/V. Ob-

viously, Λ = Λ/γ 0 2.K and Λ has no torsion.

Assume, by induction, that the Λ-module Er has a finite number of Λ-generators u\j , . . . , ^ / r '

and there exists a number ΝΓ such that Er = Er ® ^Λ r, where Εr is a Λ^ -module with the finite

number of generators u ^ , . . . , u\r), above. Consider dr(,Ujir) )= Σ k^uk

ir), where λ^° G Λ. Let

dim λ, . ( r ) < /Vr for all k, j. Set Nr+1 = Max (/Vr, Nr). Then λ^,(Γ) 6 Λ Λ By virtue of the Noeth-

erian property of the ring ANr t , the module Η (Er ® zAN

r

r

+l,dr) is finitely generated, where ANr

r + 1

is generated by polynomial generators of dimension Nr < k < Nr + i and ANr

Γ+ι <8> χΛ r + 1 = Λ Γ.

Since N

H(Er, dr) = Er+i = Η (Er ®zAN *, dr) = H{Er ®zAN

r

r

+i ® AN r+i, dr)

I+1 N

if we set ΕΓ + ι = Η (ΕΓ ® Λ ^ Γ

Γ + 1 , dr), then £ r + 1 is a finitely generated A/vr + i-module, and Er + l

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 anni-

hilate the generator. From this and the lemma it follows that Ext l'J (U* (A2n)) = 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) = Ω 0 be the unoriented cobordism ring, 0\P) = ill

Q.

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 all 0-bundles.

As usual, the c l a s s e s Wt define c l a s s e s Wω for al l ω = (α,, . . . , a s ) such that Wι = W^ , ) .

In O-theory there is defined the Thom isomorphism φ : 0* (X) — 0* (Μζ, * ) , where Μξ is the Thorn com-

844 S. P. NOVIKOV

plex of ξ. Let X = B0n, Μξ = M0n. Let u = φ (I) € 0*(M0n). We define operations

f, L) ->- Oi+d^{K, L)

by setting S q ' » = <£ (IFJ, where ίτω € 0* (B0n).

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 Sqw 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 SqA(*) = 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 SA^a^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 com-

position and forms a subalgebra isomorphic to the Steenrod algebra A C S C A .

How does one compute a composition of the form Sq" 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 = ~r

x, 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: Ω0->Ω0,


where (Χ, ξ) - (Χ χ Μ, ξ κ (~τΜ)) and (Μ,-τΜ) € Ωο represents α €Ω0.

In particular, we have the formula

w:-a= s w:t(a)-w:2,ω=(ωι, ω?)

where α εΩ 0 , ^ ( α ) 6 ί 1 ο .

It turns out that the following formula holds:

S q * . a =

where a € Λ = U,Q.

We also have a diagonal

where Δ (α) = α (8) 1 = 1 <8> α, and ASq01 = 2 δ(ϊω' ® 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± = y(Wi),

Δ;(α) = φ(/* 2 ), where h2 = 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

( ^Ci^- ^CWcAo*(MSO)-+0)=C,

where Co = A (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 ] andWall [23].

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 C Ωο where Ω5Ο/2Ω5Ο = Kerd* ft KerA* by defini-

tion of the complex C;

3) E x t ' ' ^ + S = 0, for s jt 4k;

4) Ext1 ' ^+ 4 / t (0* (MSO), 0* (P)) is isomorphic to Z2 + . . . +Z2, where the number of summands isA.

equal to the number of partitions of k into positive summands (A,, . . . , ks), ΣΑ£ = k;

5) there exists an element h0 € Ext1 '^ associated with multiplication by 2 in Εχ, such that

Ext0 '1., ->° Ext 1 ' t

r jM is an epimorphism, t = 4k, and Ext'· ' ° Ext1 + 1 ' ! t ' i s an isomorphism, i > 1.

Α Α Α Ά **"

These facts actually are trivial since

ExtAo (0* (X), 0* (7) = ExtA ( # ' (X, Z2), J?* (Y, Zi)),

and H*(MSO, Z2), as was shown by the author t 1 ? , 18] a n d b y Wall [23], 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 € Ext1'1 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 frt-bundle of dimension


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 de-

fines an operator

a: U

if we set a(X, {) = (Y a, 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, con-

nected 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 repre-

sentation 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). We set

where a = φ'1 (a), a eU* (MU) and (Ya, fa) is the element of U* (X) equal to Όα(ξ), α € 6'* (Βί/), andΓΜ 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)] · The only thing that remains

to be verified is that ο is a representation of the ring A 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 wer\_,

shall use the geometric meaning of the transformation a = φ' α*φ and hence we have given the defi-

nition 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 ca-

nonical 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],

then there arise classes σ^ζ) € U2l(X) with the following properties:

1. at = 0, { < 0; σ0 = 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:


ω=(ο>ι, ω2)

Proof. This formula can be established easily for the operation on t/* (BU) by the faithful repre-

sentation which we constructed earlier. Let (Χ, ξ) represent an element of U* (BU) and (Μ, —τΜ)

represent an element χ of Ω . We consider

5βοϊ(Χ>|) = 5ω[(Χ,|)χ(Αί,-τΜ)]= 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 is

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 * ( α δ ) = 2 ' σω,(α)σω2(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, = . . . = tn + 1 = DCP"· ' € i / 2 (Z) . Therefore σ ω * [CP n ] = λωϋ 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? x . . . x CP™-1* BUn-^MUn

and 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 have the following properties:

1 . If a € M a p (X, MUJ CU2(X), then S ( k ) a = a k + 1 and S w ( a ) = 0 if ω φ (k).

2 . So (αβ) = 2 &>. (α) ^ω2 (β) for all α, β G U*.

ω=(α>ΐ, ω2)

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 SM -Sa is a linear combination of operations of the form. Sa

with integral coefficients, so that an additive basis for the ring S consists 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. By definition, we have: it = ]*φ(1)

£ U2(CP™) andS a ) ( I i ) = /*S a ,0(l) = /*u'c + 1 (if ω = (A)) and σω (ξ) = 0, if ω ^ (k), since σ, = 0, i > 2,

for U1 -bundles ζ. 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 (u) = Σλ ω δ ω (κ) and then use property 3. Whence it will

follow for large η that Ξω o S ^ = Σ λ ω 5 ω .

The lemma is proved.

Further, we note the obvious circumstance: An additive topological basis of the ring A has the

form XiSu,, where %i is an additive homogeneous basis for U*(P), Ul(P) = Ω[/.


The topology of A is defined by a filtration. This means that the finite linear combinations of

the form XA^S^ are dense in A^ and the completion coincides with A1 , 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 im*; = — 1i. The ring S is

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 <8> S, where

Δ (5 ω ) = Σ ^ω, ® 5ω,·(ωι, ω 2 )=ω

Since MU is a multiplicative spectrum MU ® MU MU, the ring /ί has a "diagonal"

Δ: ^ - > ^ υ ® Λ 4 υ ,

where Δ ( 5 ω ) = 2 ^ω, ® Sa>, and «α Θ i = α Θ zi> = %(o ® 6) for « € fly = Λ. The Kiinnethω=(ω!, ω2)

formula for Ky, K2 GO {complexes without torsion! has the 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 has 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-^a* .

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 lass σι.

1. Let Δ ( £ ΐ ( jC2) C A be the cohomology operation such that

σι)Λιν(σι)Λ'] e U'(MU),

where σ, € U2(BU), y.U'^U3.

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 ct (f, ) = -c,^ (f,) = - c, ( | j = c, (f).

Let

852 S. P. NOVIKOV

be the self-intersection in U* (X) with normal bundle

** (^ + · ; · + . ξ ι + ίι + y + . i i ) = w>kL hi

where i: Ykl> kl -> X.

WesetA(klfk2)(X, £)=jYikl. ft2), i * ( i + ))·

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 classes and operations X(kli k2)· J u s t a s w a s t n e case for the operations &(klt k2) and

classes γ{ο% ' ) y(— σ,) 2, the operations X(k k ) and the classes corresponding to them will be de-

fined for a bundle ξ only as functions of ο^ξ) or of γ(σι (ξ)). We define these classes for one-dimen-

sional bundles ζ over CPn.

We consider the projectivization Ρ(ξ + k)^CPn, where k is the trivial Α-plane bundle.

It is obvious that τ{Ρ(ξ+ h)) = p* r(CPn) + 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 kz) such that rr'(ki, fc2) = n where

r is the realification of a complex bundle.

Ρίξ+k) has the induced t/-structure p* r(CPn) + r\ki> ki). We denote the result by Ρ^·^\ξ

+ k). We denote the pair (P(k>- ^ (ξ + k), p) € U*{CPn) by 0 χ ( Α ι , ki), where X ( k l , 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 is 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 is represented by S(/-manifolds, Δ (ki> ki

= 0 by definition; since * is a faithful representation of the ring A by virtue of Lemma 3.1,

^•(.ki, k2) °d = 0, where d = Δ ( ι > 0 ) .


The equations Δ* Ψ* = 1, θ*Ψ* = 0 were proved by direct calculation in L ^ J . Hence Δ Ψ = 1

in AU. 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:

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^, b k] = (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 U n-bundle ζ there is a cohomology operation yn., € A such that σλ(

= Yn-ι ( σ η ( Ό ) > where λ_, = Σ( — 1)'Λ' and the Λ1 are the exterior powers.

b) If ul, . . . , un G U2(X) are elements in the subset Imy = Map (X, Ml)\) C U2(X), then we have theequation

Y n _ l ( l i i . . . Un) = Υ ι ( « 1 · γ ΐ ( Μ 2 · γ ι ( . . . · ν ι ( κ η _ ι · Ι Ι η ) ) · · · ) .

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) l ies 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 yn., = y1(u1 . . . y ^ u ^ O . . .) on the element u = ul . . . un = 0(1) € U* (MU„). The opera-

tion γη.ί is well defined υα.οάχω8ω, where ω = (klt . . . , As), 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 U-

cobordism 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(^i, 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"21, 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= 2-^(1 D-i»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/iiA7) has fil-

tration 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

2 5 " ^ '>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

a = "y. So,, a(u) = ; for ah = 3 ! 5 ω , a(u) = u (1 + uk).^ ί — u 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 — Ω";

fc Μ ® . . . θ α3) Ψ[/(Μ) = — (A times), where u € U2(CP ) is the canonical element and

k

composition in Map(,¥, Λ/ί/χ) C U2(X).Ik

- υLemma 5.8. a) The series xH.,(u)has the form

x ' k /<•

b) Ψ* * (x) = kix, χ e A~zi == υ-Μ(Ρ) =

1= —f(u,f(u, ...J

A"

856 S. P. NOVIKOV

j \ XI) k ψ I __ m kl __ xu I m k .a> U U ~ U ~ U Ύ 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 con-

sider only those π for which π(χγ) = π (χ) n(y) and π{ιι) = ^XiU1 e U* (CP°°). Let

(ιXi = [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 Δχ C Λ y ® (? be the "operator of division by y," which has the following proper-

ties:

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^ = Oy, Ψχ =Ψ y ,andΦ y oΨ y = 0.

Moreover, the collection of projectors π£ = JlAl

y — 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) = (1—4>i)*yj for / > i, the collection of elements

yk is a system of polynomial generators.

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

K e r j t / = U

In particular, 1— Σττ;· and ττ; + 1 = y^jAy.. Hence/

iTj (U ® ζ*) are isomorphic.

= ?r;-(y^) for all % e ί/ , and all theories


The projector na = 1 —y,A y [ has 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

^o*(y ;) =Yj for ; 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.

s e t

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

y = 2 J

For the generators Ji €

Hence

and

we have | λ | = 1 for i ^ p' — l and | λ | = ρ for i = p ' — 1 for any prime p .

v{ = Σ yiq~isa,...,i

q-1

It is easy to see that for i + 1 Φ ρ' for given p, A y . € /I ® Z*?p> i ° r i + 1 = p ; and ρ > 2, Δ

€ /I <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 all

numbers i j. ^ p ; — 1 in the natural order, i\ < t2 < . . . < 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 as 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, sinceφί: = 1 -Jik^Yi,' a n d ^y,, commutes with yn j < ih.

We consider the operator ΦΛ.Δ χ Φ Λ = Δ^.,. Since Φ^ is multiplicative, Δ ^ is the operator of

division by Φ^ί^.,Φ^. Hence in the cohomology theory Φ .(ί/*) the operator Δ^.., has all 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 aft*i • ^ - ^ ' ^ ' ^ h ' l k - i A; "i

complete system of orthogonal projectors.Thus, - yl f c .,

Φ Λ = Φ Λ (1 - y I/ £ i

h and = Φ4-ι, while

h e r e Φ λ = 1 - y i A . A y .^. If <t>s = 1 _ y ^ A y .^, t h e n we s e t :

where ^, or:

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,

b) φΐ*] e= 4υ ® ζ Qp.

The collection of Φ' with A -> °° is such that Φ * is independent of k when it operates on Ω

and hence the sequence Φ as 4 -» », or the series 2 (φ^+'Ι — φ Μ ) = φ defines a projector

Φ € /I ® 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) UP t o a

shift of grading (suspension).

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), Ζ ρ) = 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 homo-

morphism Φ : Λ -> Λ annihilates all polynomial generators of the ring A = U (Ρ) = Ω(/ of dimension

different from p' — l.

b) Γλβ theory U <£) zQp decomposes into a direct sum of theories of the form Μ ( p ) = [/ p arao?

tAeir suspensions.

§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.


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( , IT(P))= ExtAu( , Λ ) .

2. Let X = CPa = (EkCPn) € 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. Xl

k»> = S2n + l/Zk =(EkS2rl+VZk)eS, X2

in) = RP2n + i . U*(X) has two generators u d U\X{

k

n)),

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 dis-

tinguished 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 (ya 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 € φ](γσι).

In particular, au = 0 for a = Δ(^ 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

eAv.

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 mod-

ule 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 anni-

hilator 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 (J(U (MSU), A). In

particular, it turns out that d{ = 0 for i ^ 3, d3 Φ 0, and E^ = E\' = 0 for i > 3-

For the calculation of Ext*j \j we consider the complex of A -modulese d d

C = {U*{MSU)+-C0+-Cl*-...<-Ci...),

where the generators are denoted by ut € C t for i > 0 and vt € C ; for i > 1, Co = A and C ,· = A + A

for i > 1. We set rf(u£) = du^ and d(vt) = Au^. 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, A) = Ων, we

obtain the complex

d* d* d*

HomJXC, Λ) =

where d* = d* + Δ*:Ω[/^Ωί/ + ily.


Since Δ* is an epimorphism, the complex Horn* Ay(C, Λ) reduces to the following:

8· β* a*

where W = KerA* Cfly.

From this we deduce the following assertion.

Lemma 7.1. a) For all s > 1, we have isomorphisms

X (£/'(MSU),A) = Ht-2s(W, d').

b) ExtlV (IT (MSU), A) = Ker d* Π Ker Δ* c Ώ ν .

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 = [CP 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 SU-

manifolds the A -genus is even (see L^OJ).

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 · &3 —>~ίί3 ,

where d^K) = h\ We have

Lemma 7.2. // a. € Εξ· q for ρ > 3, and d3 (a) = 0, then a = di (β). Hence £ / · q = 0 for ρ > 3, and

V = F

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 £3

P'* = Σ ExtjV (£

862 S. P. NOVIKOV

is the ideal generated by the element A, we have Εξ' q = 0 for ρ > 3-

From dimensional considerations it follows that ΕΛ = E^ .

Since E*J = £0^* + Eli* + Ell* i s associated with Qsu = π, (MSU), and E^* = hEaJ\ Eli*

= A£oo' = A2Z?So we obtain

Corollary 7.1. a) flj* + 1 = AOj* ; b) Α2Ω2£, = ΤΟΓΩ2/^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 s singled out within the intersection Kerd* f] kerA*

iy setting equal to zero a certain collection of linear forms mod 2, generated by the homomorphism

d3: (Ker d* f] Ker Δ * ) * = Es 2h-+ Εξ12h+2

= A3(Ker 5* Π KerA*)2fe-4 = Η ^ Ά ~ * = 2

b) ΓΑβ group Ext1 = ff(IF, (?*) is isomorphic to the direct sum Ως^ + Ωο Τ5, orerf iAis isomor-

phism comes from the differential d3:.

where Ext1'2^1 = Z?^'2'' = J ' 2 * 3 ^ ^ J 1 = Ker rf3, 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) Ext 1 ' , 8 * 4 6 = Κ E x t 1 · ' * * 2 . w h e r e K € Ext 0 · ? ,^* (MSU), A).

b) d3 (Ext°'j) = 0 for i ^ 8k+ 4, and d3 (Ext°''uk + " ) = Ext 3-'(J

h + 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 °) = 0. In

fact, by the induction hypothesis on the groups Ext3'?, the differential d3 is a monomorphism.


Hence Ext 0 · ' * 0 ^ 0.A u

We now consider d,(K Ext"·8^0 ) = /i'Ext0'8^0 . We see that d3(K Ext0 '8^0 ) is an epimorphism onto

Ext3,eJc0 + 6_ Whence parts (a) and (b) of the lemma follow; on the group Ext3>8'c + 6 the differential is

trivial, and on the group Ext3>8fe + 2 D Ket d, = 0.


Thus, we obtain

Corollary 7.3. a) The image Ω,ί../Tor COy coincides with Ker d* f| KerA* for i £ 84 + 4.

b) For i = 8k + 4 the image Ω ^ ^ / Τ Ο Γ C Ω8[/ί + 4 is picked out precisely by the requirement of the

"Riemann-Roch 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 classes" π^ € kO* [X\ are introduced in AO-theory. Consider the classes π2ι 6 kO (X);

let TT2I = /<;. Now consider the numbers ch(cKti . . . cKik)A(X) [X] for Χ €Ω 8 ^/ΤΟΓ C fl'y. These num-

bers are different from zero mod 2 if and only if hX Φ 0 in Ω8^1. 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 ni(kn)

= 0, i < n, and Ω2η1ί2η = BU χ Ζ. The spectrum k is such that the cohomology module H*(k, Z2) 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 -> k2n_2 by virtue of the Bott periodicity £t2k2n = k2n.2,

and k2n is a connective fiber of BU. 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 [2J 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: T2n+2j(k2n) -» "i n + 2 7( 2 n ) * s multi-

plication by kn + >.

Let α η

: = = ^ λ ^ (Α;ηΨΛ), where the A are integers, be an unstable cohomology operationft

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.

hDefinition 8.2. If the stable operation a has a zero of order q in the sense that a* ; = 0 for / < q,

then we also call b = (x'"a) a stable operation, where a = xqb, b :k\X)^ki+2q(X), 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 peri-

odicity 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 opera-

tions 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 δ0 = 1. Let y/ί* = 0 and y/*^ be multiplication by 2. By definition, we shall take

Ύ iJ = 0 for j < i and y.* to be multiplication by a number γ ι which is a linear combination

Yi = _/j μ kn+i, 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^ kn+i

kunder the conditions:

2 μΛ1>Αη+ί = 0, } < i.h


We consider the operation din — 2 .uft (knWk). 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 well-

defined.

Consider the operations ain for re -> °o; we shall successively construct the δ ; from them. We have

aon = 1; let bmn = aln + κια2η + . . . +Kmamn be linear combinations such that the homomorphisms

(fcmn)« for / < m <3i re are multiplications by integers y;,/, where 0 < y1>; < y ;. Clearly the numbers

y1>; are uniquely defined. Let m -> °°, re -. oo; then in the limit, the sequence (bmn) gives an operation

which we denote by yt = Λ^δ,. It is uniquely defined by the properties that y, * = 0, Υι*ι) = 2, and

o<Yl.in<Yi, yJ^n,,.

The operations γ ι are constructed in a similar fashion, and are uniquely determined by the con-

ditions Yiin = 0, / < i, y#> = Yh and 0 < γ ^ < yk for k > i.

We exhibit a table of the integers y; *< ; ) = y^ in low dimensions:

024

To

111

Ti

020

T2

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 opera-

tions 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 γiiq) = (xl8i)lq^ for all i > 0, for a

fixed integer q, coincides precisely with the greatest common divisor of the numbers k (kq — 1) for all

k. There exist operations a ^ j n G Β „, such that a^,,* = kn + > for j < /(re), where /(re) -» t» as re -» oo.

The proof of this consists of the fact that the operations %'δ; = y£ are obtained as linear com-

binations 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

described above for constructing {χι$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: let re be sufficiently

large that y . ( ^ = 0 for / < i and y . ( ^ = y for i < f(n), where /(re) -> °» as re -> °° 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 = 1i^ixl8i + bf(n) if ο = (αη) €βψ. Since the greatest common di-

visor 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

the basis (xl8j) = (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 ^ψ-

module 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 )

= ((v. + 1)A-1)' and Ψ(χγ) = xHk(x)xVk(y). The elements of k*(MUn) have the form Σλ;> sxsds, where

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

b2liesinB^ ifx^b.eB^. Let aln = ^ λ^knWh and a2n = Σ μ^^ΨΚ We considero

) (h h

iusing kq2xq2XVk = xHkx'12}. We shall assume that re is very large, re -> oo, qi and q2 are fixed. We set

m = η — q2. Then

2ΑΛ


where A^m) = λ^η ) and μΙη) = kq μ^ • Clearly, a s m » 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) -> °c (as η -> °°) with the operations (Α"Ψ'ε) in the sense that a-kpn,* = kn + > for ; < /(re).

This remark allows us to use (up to any dimension) the ring Βψ as if it were the ring generated

by (ρ"Ψρ), with ρ prime, and by χ e Bk

w , where ίρηΨρ)χ = ρχ(ρηΨρ) and yp = (ρ"Ψρ) are polynomial

generators. Thus, a (topological) basis here is xkP(y2, y3, . . .), where Ρ is a polynomial.

We consider the βψ -module k*(P). We have

Lemma 8.3. The torsion part of the group Ext1'£'(k* (P), k* (P)) is a cyclic group, whose order

is equal to ip"(p l—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^ = Co, β ^ _ p C C l ;

respectively.

We consider the complex Honi j t (C, k*(P)).

Let ht € Horn21 k (Co, k*(P)) be^ lemen t s such that hL{u) = x\t), and ht

{p) C Hom^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)^i · Thus, i s a 1-cocycle for the operator d*.

0 » " ( ρ * ΐ ) >0 » ( ρ ΐ ) > , d{hi)Since Hom*( , k*(P)) is a free abelian group, the element . —: , is the unique element

\P \Pl 1 ) / P

of finite order equal to dt in the group Ext1^21 (k*(P), k*(P)), dt = ίρ^ρ'-Ι)!,,, η -^ «..


Note that the computation of ExtA (k*(P), k*(P)) presents no difficulties, since the moduleψ

k*(P) has a βψ-free resolution which coincides with the complex for the polynomial algebra Z[y2, . . . ,yp, . . . ] , as long as the operator χ acts freely on k*(P), B^, .

We have

Theorem 8.1. The groups Ext l'*l(k* (P), k* (P)) are cyclic groups of order dt = \pn(pl— l ) i p , where

η is large. ™

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

for s > 0. Hence the groups Extjj'/,21 are all torsion. We consider the resolutionΨ

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) = xJ(i), then

Thus, if <i*Ay = Σ/ί -ίΊ, where A^/^i) = xhi(t), then

where AJ- (νέ) = x'(t) and the numbers μγ 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

\'^\i. Since the elements (xlSt) give a system of relations in k* (P) over the ring Βψ C Α

the same integers μ/ give the torsion part of Ext'n2/, since the complex C over Α-ψ is a segment ofψ

a Βψ -resolution of the module k*(P). By virtue of the lemma, we get the required result. The theorem

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 essent ial ly a straightforward consequence of the result of L22J

concerning the fact that the Riemann-Roch theorem on the integrality of the number ch ζΤ{Χ) ιΧ\ gives


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 proper-

ties of the Thom isomorphism in K-theory, this assertion is equivalent to the following: for any in-

divisible element α € n*(MU), there exists ξ 6 K"(MU) such that (ch£ Ha) = 1, where Η: π* -» //*

is the Hurewicz homomorphism. Let β € Horn*. k (k*(MU), k* (P)); then the number (ch<f, /3) is alsoψ

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\ . rτ* i v \ . Λ—*• U (Λ-%) —>U \Λ.) —^(7 \A-l)—*• "

I)) v h i ν I))

0—» Horn* ft —> Horn* Λ —» Horn* h —>· E x t 1 ' ! (k* (MU), ft*(X2))." I B · " ψ * - -VL·* "ΪΙΛ

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))arenon-Α"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 coho-

mology 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.

In the theory kO* it is possible by analogy with the theory k* to introduce operations (knX\!k) and

their combinations α = (on), 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 com-

posed, 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 'such 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 is a mapping, then a necessary and sufficient condition for the complex Dn + k+1

UfS" to belong to B' is that f* = 0, f :W*(Sn) -» W* (Sn + k).

In the category Β D B' the operation of the ring BL is well-defined, the latter being α priori an

extension Βψ -» 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 Ext^Ioψ "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

Exts^o (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 : ExtB^° (Λο, 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.

There is defined an element h € Ext0'1 n (AQ, AQ) such that 2h = 0, h3 = 0, while multiplicationnku

by Λ ν

0,8h+l h 0,8ft+2 l,s l,s+l

ExtB

f eo - ^ 0

is a monomorphism for s = 8k, 8k + 1.


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

image of qv ο J, where J -n^SO) -> 77*(Sn), and is nontrivial in dimensions (1, 4k), (l,8k+ 1), (1, 8k + 2).

§9. Relations between different cohomology theories. Generalized Hopf invariant.

i/-cobordism, Α-theory, Zp*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). ThetlnS 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

· C υ κ2)

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 elementg

?i(g)eExtA* (Χ*(Ki),X*(K2)).

We thus obtain a mapping

q± : Ker ςτ0

(Β) ->- E x t i f (Χ* ί^), X* (K2)),

where qa:Hom* (Klt K2) -. Horn* Α" (Λ7* (K2), A:*(K1)), K,, K2 € β and g 6 Kerq(

0

B\ provided CK,

U K2 € Β. This map is "generalized Hopf invariant."S -1-

General problem: which elements of Ext ' '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 in-

variant qiB)(g) € Ext1 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 theexamples 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 Ext1· {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(

0

Y) • } = 0.

If Κ = EL, then in this case the computation can also be carried out by means of Ext γ (Υ (Ρ),

Υ (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.


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 augmenta-

tion functor

ν: Χ*^Η*(Υ,π)

and hence for any group G a functor

yG: X'-+H'( , π® G).

For example, for G = Zp we have vp:X — Η ( , π ® Z p ) . In the cases of interest to us, π = Ζ and

rr<8)Zp = Z p .

3. The Riemann-Roch functor. Let X = U and 7 = A: ; we consider the Atiyah-Hirzebruch-

Grothendieck element λ[η) € K°(MUn). 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 > a n 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- CY^aCx

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, a,V)) = R*GN(M), where R* = ^ Rq ar»d GN =Β 4

Hom^V ( , a.N)°a. is the composite functor, RqG is the g-th right derived functor. There is defined aΒ

874 S. P. NOVIKOV

natural homomorphism

rq: Ext^x {M, N) -> R"GN (M), r = 2 rq,AB q

and homomorphisms

Pi': RqGN (M) -> Hq* (Hornby (CY, aiV))AB

(y(aAf, aTV),AB

where Ker /82* = 0.We have the composite map

a= (tir^W. E1(a)-*Ext1-7(Y*(Kt), 7·(ΛΓχ)),AB

where

£x (a) c ExtU (X* (tf2), X* (^)),AB

Ex (a) = Γ^οβΓ'βϊ (ExtU (Γ* (ΛΓΟ. i " (^i))·AB

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' q

— 2 RpGq,N(M), which converges to Ext** (aM, a/V), and GQi N (M) = Ext?. * ( , e\')°c. From this

P, 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 λ : U -> k a n d v p : U - > / / ( , Z p ) are algebraic i n t h e s u b -

category 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

^i ( , λ/ν)°λ, λΝ = k*(Kj.

d) The functor vp is such that RqG^(M) - E x t ^ , » , ^»( i/ p M, i'pN).


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 Λρ = 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 homo-

morphism 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^'y1 (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.

876 S. P. NOVIKOV

Proof. Since the groups Ext1.'!1 (A (P), k (P)) by virtue of the theorem are cyclic of the assertedΨ

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-

morphism a. = (β*'1 · β* · rr).

The proof of this lemma follows immediately from the fact that by construction of the generalized

Hopf invariants qlx and q^Y we can compute both quantities qfy (a) and qfy (a) for any

a€ Ker ^ ' C K e r q0(p.

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 -. ΥAB

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'' =B Λ β

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= 5ω(ω), dim ω > 0. We con-ω

sider the differential

ω>0

where

d* (x) = 2 Υ*ω (x), x ^ ®u = Hom2l« (Co, Λ)


and

( C \ ) > Ω

where σ%{χ) ίαω'] = 0 if ω Φ ω ', and σ&Ο) luM] = 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(χ{) = 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

-» Ext1'^ is monomorphic and \kn{ki — \)\]c = 2, i'= 1 (mod 2), we have

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 M21, 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 easi ly 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

AExt1>2i C Ext1·2,' 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 Ext1 '8^ (Λ, Λ) are isomorphic to the groups Ext1'8^ (k* (P), k* (P)).

In the case a[ = 1 there arises an uncertainty: do the groups Ext1^8 + 4 coincide with the groups

Ext*'^ + 4 or do they have index 2 in them?

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 S2/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 denom-

inator 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 1,2

d3(uh) = h3ExW (Λ,Λ), fee ExtA" (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, k ^ 0,

Extfifco (kO*(P),kO*(P))=0, i=#8ft + l, 8ft + 2;

b) E r t i $ (ftO* (P), ftO" (P)) = Z{h«(ft*-i»fe + • • ·, η -» oo,

Exti^o+< (ftO* (P), ftO* (P)) = Z2 + . . .

for f = 1,2;

c) the homomorphism q0 :π$(Ρ) -> Horn ^ 0 (ΛΟ (Ρ), kO (Ρ)) is an epimorphism (result of Brown-

Peterson-Anderson L J); ψ

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); sinceψ

l,8ft+i l,8ft o,lExts*° = hlExtfi*° (t = 1,2), Α ε ExtBfco,

the required fact follows for the groups Ext1'*Aft(Ao,Ao).

We now consider the complexification c, defining homomorphisms c, "c ':

Exthko(Ao, Λο) — Ext^%(A, Λ) = Ext1;** ,

^ i 0 (Λο, Λο) — [

Since in the groups kc(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 Ext1'^ (Λ, Λ) has index 2 for I = 8k + 4, index 1 for I = 8k and is equal to zero for ι Φ 8k,Αψ,

8k + 4, since Im^ = Im'c' .

Consider the groups πη + *k-i (Sn)and the Hopf invariants in kO - and k -theories. These invariants

are always defined since Ext0'4 = 0. We have thus the

Conclusion. The image of the Hopf invariant

1,4ft

quh: nn+ik-i(S»)^ExtA^(kt(P),

h a s i n d e x 2 for k = 2l + 1 and i n d e x 1 for k = 2l. Moreover , t h e image q1>k(1Tn + 4k-i(Sn)) c o i n c i d e s wi th

the image quk • Ιπ^-ι (SO).


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 Ext1'4,, (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 .Au Au

3. We now consider the functor vp : U* - H* ( , Zp) and the corresponding Hopf invariant

Jtn+i_i (Sn), η -> oo, i > 1

Γ^ β ( Ζ ρ , Zp) = E x t ^ ( Z p , Z p ) .

Since Ext1 '1 (2 D , ZD) = Ext/i(ZD ZD) for i > 1, this becomes the usual Hopf invariant. SinceΑ/ βΑ + Αβ μ ^ HP r

the homomorphism

qw: / j t 8 f t - i ( ^ O ) - ^ E x t ^ (ΛΛΛ)

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 ele-

ment 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 homo-

morphism

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 -^ExtA/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))^Ext'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 €. Ext1·^ ({/* (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:

a) d3 (hKym) = h4ym ^ 0,

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) = Z2 are cycles for all Adams differentials and

belong to the image of the Hopf invariant

qiu- nn+sk+i(Sn)-Êxtû (Λ,Λ), η->-oo.

b) The groups Ext1'^, + 6(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 Ext1'^ (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

ExtÛ (Α, Λ)-

t

Ά (Zp, Zp)

1; U

1

yk(

Extl:h*o{kO'(P),kO'(P))Β γ

1i,kO

. (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 Ext1

υ (Λ, Λ) - Ext1

y (U* (MSU), Λ) was studied.

3) It was established which elements of all these groups Ext1 are realized as images of the Hopf

invariant qx. In particular, for the groups Ext 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)


is nontrivial; it can be shown that d,(El

2'2t) = h3E\·"'* for t = 4k + 2 (k > 1) and t = 4k + 3 (k > 0)

(see §11).

4) The nonexistence of elements with classical Hopf invariant 1 is a consequence of the fact

that "ΰ2 (Ext1'2J) = 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 S®?Qp

Earlier, in § 5 , it was proved that in the algebra A1 <S> zQp there exists a projector Φ €. A <£)zQp

such that Φ(χ, γ) = Φ(%)Φ(ν) and ΙπιΦ* C Λ is the ring of polynomials in generators y,, . . . , y _ _ _ ;

dim γι = 2p'— 2, where the y ; are polynomial generators of the ring Λ = Ω(/<8> zQp such that the num-

bers σ* (y) € <?p are divisible by ρ and au {jCi = P> k = p 1 — 1. Moreover, a complete system of orthog-

onal projectors Φ ( 0 was constructed, Σ Φ ( ι ) = 1, φ ( ί ) · Φ < ' > = 0, ι'Φ j , where the Φ ( ί ) (U* <g> zQp) are

isomorphic theories up to shift of dimensions. Hence, in the category S ®zQp the spectrum ML·' is

equal to the sum MU Λ: ^ j Eld^Mv , where ω is not p-adic j i .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 /lp = Φ · -1 · Φ,q ~ 1

then we have:

1) \ U ®zQp = ^'^ (Λρ) i s the appropriately graded ring of infinite matrices of the form ( α ω . ω .),αω -ω- ε /Ι ρ , ωι not p-adic and dim (αω , ω .) = 2(/(ω;) — 2ί/(ω;) + dim αΜ.ω. I i .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 (Ul(K), 6'*(L)), where U* = Φ(ί/*®ζ<?Ρ) is the theoryη

defined by the spectrum Mp.

3) The Adams spectral sequences (Er®Qp, dT ®QP) in {/-theory and (Er, dr) in i/p-theory

coincide. These 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 suf-

ficient 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

in the ring Φ ^ ^ Φ έ it suffices to consider only Φ, · Φ · 5 ω · Φ · Φ; for ω = ( ρ Ρ ι - 1, . . . , pqs - 1), where

all <7;- < i.

Additive bases for the rings Ap and Ap>i:

a ) Ap = (Λρ ·δ ω ) Λ ,where ω is p-adic and / \ denotes completion (by formal series).

t>) ΛΡι1 = (ΑΡι1.8ω)Λ, ω = ( ρ ' ι - 1 , · · · , pis-D,jk <i.

We consider the operations e i ; ^ = S, i. ι (^ times), regarded as elements of the ring

APil, i.e., eiik = Φ ^ α , Φ Φ ; € AptL. Clearly, we have:

1) Δ(βί,&)= 2j eis®6i:i (the projectors Φ; and Φ preserve the diagonal);h=l+s

2)e? > t (A p > i . 1 ) = 0;

3) e j i (γι) = ρ, where yi G Ap>j C Λρ 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 spanned by &i and the operator of multiplication by the gen-

erator yit i.e., Dt = Cp [ y j &t.

We have the following

Lemma 1 1 . 1 . a) The subring fej commutes with all operators of multiplication A.piimi C Λ ρ are</ a//

operators Φ^Φδα,ΦΦ; /or οΖί ω = ( ρ ; Ί - 1, . . . , p / s - 1), z^Aere j k <i - I.

b) /« ί/te rireg &j we have the relations

f +ei,h • 6i,s = I ) · β j , ; l + S T

\ s /

ei,h-yqt= 2 el.

s-\-m=i

where e*m{yi)= ( Q ) p™yV~™.

c) The ring APii.i is obtained from the ring APti by discarding the polynomial generator y^ and

then factoring the remaining subring BPli C .4p > !: by the ideal spanned by the central subalgebra fe;

of ΒPj;, 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 lp,;-i in the following way (in two steps):

Step 1. Without altering the "ring of scalars" Apii.lt we make a central extension of fe; + 1 by

AP.i-·


with Q>i acting trivially on Λ ρ ^ ^ .

Step 2. We adjoin to the ring of scalars APtiml a polynomial generator ji of dimension p' — 1,

setting eiik(yj) = (pp/cyq-Zc with all the consequences derived from this.

The 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;5ωΦ; 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 on the generators yi

(and even only of the Pp ).

We now consider the ring Dt, operating on the module Qp LyJ, and the groups Extp f i (^ p LyJ,

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 ® Λ Ρ ΐ έ . , such

that:

1 ) λ ( % ® μ ) = Λ : ® λ μ , λ , μ € Λ ρ > ί Μ C A p > i . , -

2) if eitW = Φ £ 5 ω Φ Ι ,

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 .^(^Ip.i-i, Γ··«®Λρ,(_ι)

= H o m A j i i M (AP,i-i,AP,t^)

Proof . P a r t (b) i s o b v i o u s . T o c o n s t r u c t the a c t i o n o f . 4 p j . j o n Γ 5 ® A p > 1 . , we n o t e t h a t the r ing

β ; a c t s on Λ ρ > ; = Qp [γ] ® A p > 1 . 1 n a t u r a l l y , w h i l e the 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 fo l lows the

n a t u r a l a c t i o n of t h e factor-r ing APti.l on t h e g r o u p s

Ext D . (AP,ir Qp [Vi]) = Ext D . (Qp [y], QP [y]) ® Λρ.ί-ι,

w h e r e Dt = Qp [yi · &i. It i s now e a s y to 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 spectral sequence (Er, dr), where:

a) Ea> is associated with Ext/ι (Λ Ρ ; ί , A P j i ) ;

b) Ev'q coincides with Extp. (Λρ,,-,, Γ" ® A ^ . J , where Γ" <g> Λ ρ . ^ is a Αρ^.,-module byρ i• ι

virtue of Lemma 11.2;

c) dr:Ep

r'q -> £ P + Γ· 9 * r + ' ; - a// differentials dr preserve the dimension of elements induced by the

dimension of rings and modules;

d) E*-° = Extp

Ap^{KPti.1, KPti.x);

e ) iAe spectral sequence (Εη dr) is a spectral sequence of rings, where the multiplicative structure

is induced by the diagonal Δ of the ring 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 . We shall not give it here.

For what follows it will be useful to us to compute Ext^' (Qp [γ], Qp lyi). We note that APli=1Dl,

and the calculation of these groups gives certain information about the ring

ExtA

u (U*(P), U*{P))®ZQP.

Lemma 11.3. Let C be a bigraded differential ring over Qp, which is associative and is generated

by elements

such that:

Λ. ι 7//tjj_i —— We. ·*ΐ| " • CCfbj —~ ftjCCl

2) d(x)=phi;

3) d(hi) — O, d(hj+l)= 2 ( , )hj+i-k-hh, j ^ 1;

4) d(uv) = (du)v + (— 1)έϋ(ο?ι>) where u € C1' . Here, d is the differential in the ring C

Then the cohomology ring H (C) is canonically 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 (C ®Z2) is isomorphic to the polynomial alge-

bra Z2 [χ] ® Z2 [hl, h2, . . . , h2k) with Boks'tem homomorphism β of the following form:

2) ρ (h2k) = /i2k-i, « > 1, x c. Η · , h2k t H2 <2 -1>(L *& Δ2).


b) For ρ > 2 the ring Η (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 LyJ , Qp LyJ) is nontrivial for t = 2p(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:

1) AjX9"1 for p> 2 and all q,

2) h^'1 for q = 2 an<i </ = 1 mod 2,

3) Α , χ 9 ' 1 + 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) Zp) = Z p [x] (g) E x t ^ , Θ Ζ ρ (Zp, 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 differen-

tial d* in the complex Horn* (Qp [ y j · F, <?p t y j ) . Namely, we have:

886 S. P. NOVIKOV

1Part (d) is derived from the fact that -j^jd(xq) mod ρ is equal to Λ,χ9'1 for ρ > 2 or ρ = 2, q = 2s + 1,

and is equal to h^'1 +. h2xq'2 for ρ = 2, q = 2s.

We shall now prove part (e). Since the homomorphism β is a differential operator, it suffices to

show that Hl (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 : Ext" . (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), k^ix m ^ 0 .where ρ > 2, where

b) ah = $(h2kxm) = (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 = αϊ ,

, χ χ (2q+2l-2b (2q+2l-i)c) 02z · o 2 m = αϊ + α 2

In particular, we shall denote the element S t by h 6 Ext1'2. Hence, from (a) and (b) it follows that

025+1 · <%n = /i02g+m for all q, m.

We note that Dl =ΛΡΛ and there is defined a natural homomorphism

yd)

ExtD;# (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.

b) For all ρ > 2 the homomorphism γρ

1^ 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> and 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 with the image of the Hopf in-

variant (jrp (7r»(S™)) = q\? (/π* (SO)). For ρ > 2 the image Imy coincides with the image of the Hopf

invariant qU (π* (Sn)) = q[' (In* (SO)) in all dimensions.


In the formulation of Theorem 11.2 the calculation of the group Ext 1 '^ + 4 ®zQi l s n o t complete -

is the homomorphism y2 an epimorphism or does Imy2 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

d2 : Ε2°Λ-+ΕΪ'° « Eiti, ; i(A2,,, Λ3ι1) = 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 = Qp[yi,y2].

T h e a c t i o n of the o p e r a t i o n Φ · Ρ · Φ on Λ ρ ι and Λ ρ > 2 i s g i v e n by the f o r m u l a s :

(0, k^p, p + l,

Φ-Ρ''-Φ{χ2) = Λ ρ ) Χ ι ' k==P'

As a consequence of this we have

Lemma 11.5. The action of the operators Ρ on the generators y2 of the ring Λ ρ > 2 is given by the

following 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+

ι

ί)ρΐχι

+ι-1 for ι φ Pt

b)

S. P. NOVIKOV

We note that Pl (y2) is divisible by ρ for 1 ^ ρ and Pp (y2) is not divisible by p.

Now we can describe the action of the ring Ap>l on Γ ' ® Λ ρ ι , where Γι·ε = Ext^ ' s (Qp [ y 2 ] ,

Qp [ y 2 ] ) and Λ ρ > ι = Qp [ y j = Qp [XJ, Xl = [CPP~1]. The groups Γ1 '* were computed in Lemma 11.4,

part (c). The generator of the group E x t 1 ' ? ( p 2 " l ) ( ( ? p [y2], Qp iy2}) is obtained as d* (xq)/pflq) in the

complex Honip (F, Qp [ y 2 ] ) where F is a D2-ftee acyclic resolution of the module Qp [ y 2 ] ,

x9 € Horn* (D2, Qp [y2]) 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 Ly2J).

We set

where

by virtue of Lemma 11.5 and ak € < p Ly,J, α^ = λ yf4. From what has been said it is easy to derive

Lemma 11.6. The action of the ring APil = Di on Γ1 ® Qp [y2] 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 11.6 fo l lows e a s i l y from Lemma 11.4 and the definit ion of the generators aq = d xq /ρ ,

where xq C Horn* (F, Qp Ly 2 J) i s such that xq (1) = y2 G Qp Ly 2 J . Further, we compute Horn* (APjl,

Γ 1 ® Λ ρ 1 ) = El'1 in the spectra l s e q u e n c e (E2, d2) of Theorem 11.1, which converges to Ext^* ( Λ ρ > 2 ,

Λ ρ > 2 ) ; here ΑρΛ = D1 and Λ ρ > 1 = Qp [yj.


L e m m a 1 1 . 7 . The groups Horn*- (ΑΡι1, Γ 1 ® Λ ρ ι ) are spanned by generators K ; J 9 of dimension

2 p ' ( p 2 — 1 ) + 2q ( p — 1 ) for all i > 0 , q > 0 , where the order of the generator K j > q is p .

The proof of the lemma follows easily from Lemma 11.5 and 11.6 by direct calculation.

Since Horn* (Λρ > 1, Γ'(8).Λρ > ι) = /s"'1, our problem is to calculate d2 r i"' 1 - E2'° = Ext^ ( A P j l ,

Λρ>1), where the latter groups are computed in Lemma 11.4 and in the conclusions drawn from it.

Direct calculation proves


Lemma 11.8. The differential d2 -.El'1 -> E2

2'° of the spectral sequence (Er, dr) converging to

Ext4 (^p,2> ^p,2) i s given by the following formula:

where h ι and χ are in the notation of Lemma 11.4, and β is the Bokstem homomorphism Η (C)

— Exto (Qp C*J) described in Lemma 11.4.

From Lemma 11.8 follows the important

Corollary 11.1. a) For ρ > 2, the kernel Kerd2\E°2

l 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 ^ (ΛΡιι; Γ1 ® Λ Ρ ΐ 1 ) . * > 0.

Hence, the image of the homomorphism

; ) i l ,ι; A P i l ) - ) - E x t A p | 2 (Λ ρ, 2; Λρ, 2)

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 homo-

morphism 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) = 0 for all

s > 0, while for ρ = 2, β (h.x23 + ' ) + 0.

Comparing part (b) of Corollary 11.1 with Theorem 11.2, we obtain the following result.

Theorem 11.3. a) In all dimensions t ^ 4, the order of the cyclic group Ext1' (U* (P), U* (P))

coincides exactly with the order of the group Ext l ' l (Κ (Ρ), Κ (Ρ)), and this isomorphism is induced

by the Riemann-Roch functor λ. ψ

b) The Hopf invariant

qi: πη+,-ι^—ΕχΙ1·' (*/·(/>), V (Ρ))A

is an epimorphism for t = 8k, t = 8k + 2 and t = 4, and the image lmqi has index 2 in Ext1'' for

t = 8A: + 6, k > 0, and I = 8k + 4, k > 1.

Corollary 11.2. The generators aq of the groups Ext 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 2aq are cycles for all differentials).

S u p p l e m e n t a r y r e m a r k . 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 to p r o v e t h e formula di{a.q) = A 3 · CLQ.2

t 1 'for q = As — 1, 4s + 2, s > 1, where h = al €. Ext1 '^. In particular, for (/ = 4s + 2 this follows from

e

homomorphism, and we must have in £ „ that Α 3 α ς . 2 = 0.

the fact that hiciq.2 φ. 0 in Ext4 y, while at the same time o.q_2 is realized by the image of the J-

*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 Yi + 1) = ;1/;1

M= {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 -free modules Ν = (/V£/) is constructed.

I I l '

| dx Jd, jd,

No,o£-NOti£.No,*+- · · · ·

such that (a) < ο?2 = - rf2rf,; (b) ί-> . . . - /V[; Ίΐ /V£.l>;· -> . . . I for all / is an ^ X - f r e e acyclic resolution

of the module W;-; (c) if ζ> = 2 ^ » 3 a n d ^ = i + 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 /^-module 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 ar ises a spectral sequence of the double complex (Et

r'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 set of objects ( Z i ; ) , i > — l , / > — 1, and morphisms

Z_! _! <— Z_!_ 0 <— Z_ l j x <— · • ·

I 1 iZQ _X <—Z,, 0 <—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 - y 0 - ' . . . - y ; - ..."


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) = Nkii.

c ) T h e d i f f e r e n t i a l s d1 :Nkti -· !^k-i,i a f i d d2 :Nkti — i^k,i-i c o i n c i d e w i t h t h e n a t u r a l h o m o m o r p h i s m s

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 some d e d u c t i o n s from the p r o p e r t i e s of the geometr ic r e a l i z a t i o n of a double c o m p l e x :

1. T h e f i l t rat ion Z . 1 ( . , = Υ <- Z o M «- Zly.1 < - . . . < - Z ; _t <- . . . r e p r e s e n t s the geometr ic rea l i -

z a t i o n of the / l X - f ree r e s o l u t i o n \H* (Vo) ί2 Η* (,Υ,) «- . . . Κ

2. T h e f i l t rat ion Y «- Z.t Β | J Z O . , < - . . . U Zj ; *- . . . r e p r e s e n t s the geometr ic r e a l i z a t i o ni+j=ft-i

of t h e A - f r e e r e s o l u t i o n

where d = dl + d2.

3. T h e double complex (Z) d e f i n e s two Adams s p e c t r a l s e q u e n c e s :

a) the Adams s p e c t r a l s e q u e n c e ΕΓχ in the theory X , induced by the f i l t rat ion

y "*~ o,- i U •Z-i.o -«—...-«— U Ζ,, j -«—...

b) the spectral sequence Er of the filtration

with term £* = Ext4 ^ (A"* (Y), A* (K)) for any /( € S;

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 Zi,} = 5 . . . => U . Zu = . . . = > Z _ i l f t .i+j=fe-l i+j=h-l

c) For the second Adams spectral sequence the filtration in Er and £co is induced by the geo-

metric 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 ele-

ment 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 ( , Z p ) in the category Dp,p )

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) +ΎΡ(Χ),

where i = irj*, ίχ = L t , ζ . 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.


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.


We now consider the graded ring Λρ C Ωυ ® zQp, where Λρ= Qp [xu . . . , % , , . . . ] , dim*! = 2 p ' - 2.

The ring Λρ is a local ring: it has a unique maximal ideal m C Λρ such that Ap/m = Z p . Hence the

bigraded ring Λρ = ^j mt/mi+i is an algebra over Z p , and Λρ = Z p [h0, h^ . . ., h^ . . . ] , where h0 isρ

<—0a s s o c i a t e d with mult ip l icat ion by ρ and dim A; = ( 1 , 2p [ — 1), i .e . , hi £ m/m2. Clear ly f]ml = 0 and,

by [ 1 5 ' 1 7 ' 1 8 ] , we have:

Ap = ExU (H'(Mp,Zp),Zp).

As was established in s 11, the action of the ring Ap on Λρ = Up (P) preserves the filtration gener-

ated by the maximal ideal m. Hence it defines an action on Λρ, which is described as follows:

1) the action of Ap onA p is defined by multiplication;

2) the action of Ρ on Λρ 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 cons ider 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, such that A/./Q i s isomorphic to the quotient Α/βΛ [J Αβ and Ext/i (//* (Mp,

Z p ) ) = E x t Q ( Z p , Z p ) = A p = Z p [ k 0 , h l ? . . . , h i . . . 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) = /trM, 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 / (Λρ, Λρ') = Extiz/Q (Zp, Aj) = Exti / / Q (Zp, ExtQ

f (Zp, Zp)

(here, t is the dimension in Ap defined by the filtration Ap = mt/'mt + l ) . Moreover, if } C Dp, then for

Lp = <.'*(}•) and Μ = Η* (Υ, Ζ ρ) = L/mL we have:

a) 1/ is an /l//D-module;

894 S. P. NOVIKOV

b) there exists the identity

Extj'(L, V ) = EXI'AJ/Q (M, Ext Q

J (Z p , Zp)),

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, ExtQ(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:Λρ

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 Es

r' ' 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 con-

verges to Ext/i (M, Z p ) ;

b) since /j Λρ is associated to Λρ ', 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:Ap

r, t-τ+Ι, q

y . 75s, ', 1 Ps+L. i+r-1, q+r-l

2 ~ s,t',q&co is associated with

s+t=m

ό ' Qb) The group 2ll - οό ' Q l s associated with

ExtTup (L, ΛΡ) = 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))


we have two "dimensions": (m, q) = (s - !, q) is the "cohomological" and (s, q — t) = (s, /) is 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 (E r, dr) has all differentials

dr - 0 for r > 2. The groups

s+i=m s+t=m

are isomorphic to Ext™'g(;l/, Z p ) , where Μ == Η* (Υ, Ζρ),

( Z p , Zp) = Zp [h0, ...,hu...l d im fti = ( 1 , 2p* - 1 ) ,

and the algebra A//Q generated by the Steenrod powers Pp acts on ExtQ ( Z p , Z p ) in the following

way: Ppi(hl + i)=hh Pk(hO) = 0 fork>0,and 2

From Theorem 12.1 follows

Corollary 12.1. For any complex Υ £ Dp? where ρ > 2, there is defined an "algebraic Adams

spectral sequence" (Ε Γ dr),where Es

2-l-q = E x t ^ g (,tf, ExtQ(Z p , Z p ) ) , iAe group 2 •#!' ?' ' = E?' q

i s a s s o c i a t e d t o E x t ™ ' 9 ( 1 / , Z p ) , d T : E s

T

q -> Ef + + r - i , ? + r-i^ a n ( ^ t h e g r o u p 2 . ^ c a ' 7 i s a s -

sociated to Exts ·}, (UZtY), Ut(P)), Μ = //*(Y, Z D ) .• 4 P

We prove Theorem 1 2 . 1 . In the Steenrod algebra 4 for ρ > 2 there i s defined a s e c o n d grading —

the so-ca l l ed "type in the s e n s e of Cartan," equal to the number of occurrences 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

where τ is the type and A T» · <4 T2 C 4 T i + T 2 · 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)=

e that for Q C

r(hi) = 1, Λ, € ExtV, (Zp , Z 2 ) , and the type is an invariant of the spectral sequence (dr, Er) for r > 1.

where / — r s 0 mod 2p — 2. We note that for Q C 1, r(()r) = 1 and r(Ph) = 0. It is also obvious that

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

Exts

A'^Q (Z p , Ext^ (Z p , Zp)) the type r = ί and r(dry) = r(y) for r > 1.

This implies the isomorphism

ExCq(M,Zp)= 2 Ext^/Q(M,Extp(ZPlZp))

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" (Er, dr) of Corollary

12.1 is canonically isomorphic to the sum Έ,Ε^'1'11, where El'l'q = Ext^' i l < 7 (M, Z p ) , t is the Cartan

type, Μ = Η* {Υ, Z p) for Υ € Dp, and 2 Ex t f ' 9 (Λ/, Z p ) = Ext™'9 (M,ZP).

In this spectral sequence

Ύ . J7S, t, q 'j7 s + 1> t+r-1, 7 + r - l

and the group 2 E^''9 i s associated to Ext5 ' ' (U* (Y), Up(P)).t~q=l A Ρ

From the geometric realization of double complexes as defined above, Theorem 12.1 and Corol-

laries 12.1, 12.2, there follows

Theorem 12.2. The "algebraic Adams spectral sequence" (Er, dr) is associated to the Adams

spectral sequence (Er, dr) in Η ( , Zp)-cohomology theory for all ρ > 2 in the following sense:

1) ET-q = 2 1|· '• = Εχί%·9(Μ, Ζρ);

2) if for some γ €. El'l'q we have (/; (y) = 0 for i < k and dk(y) /- 0, then there is a y such that

φ(γ — 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;

3) t'/y'e Ε χ $ ' 9 ( Μ , Z p ) fs suc/t i/iat ^(y") = 0 for i < k and φ(ά^γ) > φ(γ) + I, then for the pro-

jection y of the element γ in Ext™ y < y ' q [M, Zp) we have the equation di(y) = 0 for i < k (we

note that for elements J £ 2 J Ext^'' 9 (Μ, Ζρ), φ (y) ^> a).

The groups Ext 1 ' ! (Up (P), Up (P)) were computed in previous sections; they are cyclic forΆ 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 Ext1 '!* ( i " O ( ' V <%) has filtration (1, k - f(Jc))>ror,

in other words, φ(α/ί) = k — f(k) in the term Ew of the 'algebraic Adams spectral sequence" (Er, dr)

for ρ > 2. Since Ext'*u (Λρ, Λρ) consists of cycles for all Adams differentials in Up-theory, dt (a/c) = 0,

i > 2, and there is an associated element ak Ε nt (Sn); we have φ(αιί) = k — f(k), iji (a^-j _ 1, i^ (a .)

< k - f(k) + 1.


Proof. As w a s shown in ^ 1 1 , the homomorphism E x t ' ( A p i , A p > 1 ) — E x t 1 y ( A p , A p ) i s an epi-

morphism for ρ > 2. For the r ing Dr and the module A p i l — Qp lxlJ the ring C -- Horn (l· , A p l ) w a s

d e t e r m i n e d ( s e e L e m m a 11.4), where φ(χ) -•• 1, φ (hi) = 0, φ (ρ) ~- 1 and d(xh)^=^ Σ f )\ / /

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'' )d(xk).

From t h i s we h a v e :

• · Γ / * \ , · , 1cp(aft) = ininy φ( . )+J + k —j — f(k) \ = k — f(k)

I \ j ! J x ι

Thus, the filtration φ of the element ak i s equal to k — f(k), s ince 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 gen-

erated 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, £ Ext^ 0 ' 2 " ' ( p - ° (Z p , Z p) and h0 £ E x t " ' 1 · 1 ^ , , Ζρ), where E x C = 2 Ext i ' " 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 pos-

sible 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

"•» ($") ®zQp a r e 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, Z2) is associated

to U2 (P) = A2. The sequence (Er, dT) converges to Ext u(i'2 (X), A2) 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, Z2), Z2),

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, Z2). We write an A//Q-

resolution 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 generators:

AosExt0·1·1, χι e Εχί1·0·2, A4 e Ext0·1·2 '4 4-1, i > 2, y^Ext 0 · 3 · 6

αίΐί/ is given by the relation hoxi = 0.

We note that the dimension of 'Ex.ts't'q in Η ( , Z2) 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 is conjugate to the generator ηω of the module

Μω, ω = (ku . . . , ks),dimua> = 8(ΣΑ;·) and νωνω = ν(ω_ ω^ by virtue of the diagonal in the module M).


The proof of Lemma 12.6 for ET follows easi ly 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), Λ2) on the bas 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 homol-

ogy 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

Ext1

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 u@v

= f(u, v) has the form

U<S>V=U-\-V-\- 2 xi,juivK

where xt- € A ' l ( t + ' ' l ) = Sl\fl + ''i'> are coefficients 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

= I C P 1 ] .

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 thatiSsO

g(u®v)=g(u)


900 S. P. NOVIKOV

Theorem (A. S. Miscenko) . The change of variables u -> g(u), where g ( u ) ^ 2J un+i,iL ι J.

hjsO '

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 deter-

mines 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' · · ·u s us+lo • • • °unr

ω = (ki,..., ks),

SWSa(Ui ...Un)= 2 « Λ . ( » 1 ·'. . »n) + a0S(k,(O)(Ui . . . 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) , fc V (ft) , i

U' = 2 j A i w . U = 2 j Μ·< U '

Obviously, S^Au' = AS(fc)u', since t\u' = u ® 1 + 1 <8) u'. Since

i i j

we have/ f~r A / ^C^ " ^ 1 / * ι / · 7 \ Λ \ i^) / /

i 3


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 jA^ = 0, /t > 1, we have

S (σ(ίι)λι + (i — &)λί_^) μ/ = 0i

for all /' > 1, and since ^j μ;· Xs = 6j, we havei

S 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* [CP 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¥k

u (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, . , . un 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 decom-

posable 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 fol-

lowing semigroups in A

*From Theorem 1 it follows easily that all operations Ψ , are well-defined over the integers on U° (X), asin /(-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 require-

ments of Theorem 1:

ψ£ (α) = q-maW£, dim a = 2m,

Jus t as earlier, a multiplicative operation a £ Αχ is 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. Let Κ = Q LfJ, Λ ^ = Λ ® z/\. We

consider for all integral values I the series Ι Ψ^ (u) £ U* (CP°°), defining the series «Ψ , (u) e L^iCP™

®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 ~ υ2) Ψ^'* :Λ-Κ

2ί - Λ"κ

2ί 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) = Z2, 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 super-

position of series ax · a2(u) takes into account the representation of A U ® Κ on Λ ®K. Hence A K 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 £ Λ2' 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 multipli-

cation 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 Λ1 = Λ*"£, Λ1 = 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-

tors of Sx (KJ) 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

to Sx (K1'1) 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), where z ; is such that dzy = 0 in S^ (K1'1) and the homology class L z ; J

6 A"* (X1*1) is represented by the element equal to <9u;, where d : X* (S[ \ / . . . ν Sl

q .) -> 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)^ASx(Y2)^Sx(Yi X Y2),

which is an isomorphism of complexes?


Now let X = U.

Conjecture. For a pair Yl, Y2, the complex Sy (Y1 χ Υ 2) 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 ^A we shall denote by

U* (Y, A), and the homology of the complex HoniQy (Sy, A) by U (Υ, A) (cohomology with coeffi-

cients 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 is a ring and there is defined a homomor-

phism λ : Ων -> Ζ [χ], such that λ (y) = Τ (γ)χΚ

3. A = Ζ [%, χ-ι], where d im* = 2, dim χ"1 = - 2 and xx'1 = 1. Here 4 is a ring, while Ω{7 acts

on A just as 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 A = Ω(/, Ζ LxJ, Ζ Lac, x~'J, Z , iAe corresponding cohomology

groups U ( , A) are isomorphic, respectively, to the cobordism theory U for A = ily, to stable

k*-theory for A = Ζ [χ], to unstable K-theory K* for A = Ζ [χ, χ·1] and to the theory H* ( , Z) for

A = Z. The homology theories U*( , A) for A = Ωΐ), Zlxi, Zlx, x~l\, Z, are isomorphic, respec-

tively, to {/*, A*, K* and //* ( , Z).

Theorem 1. Since the complex (Sy (Y), d) is a complex of free Q,y-modules, there exists a

spectral sequence with term E2 - Ext^*,, (i/* (Y), A) which converges to U (Υ, Ί ) , and there exists

a spectral sequence with term E2 - T o r * ^ (U* (Y), A) which converges to ί/*(Υ, A).

Theorem 2. Since the complexes Sy (Y)(g) Ω.,Zlx\ = S/C(Y) and Sy (Y) (gjjj Zlx, ar1 J -. S/C(Y)

are complexes of free Α-modules for A = Ζ lx\, Zlx, %"'J, and the ring Ζ [χ] is homologically one-

dimensional, we have the following universal coefficient formulas:

1 ) 0 - » - Extzi* ] (k., Ζ[*])-*• k* -»- Homzcr] (*., Ζ[«])-> 0,

2) 0-»-*. ®z[ I]Z[a:>x-1]->^.-»-Torz [* r](A;.4Z[a;,ar-1])->0,

3) 0-*-Λwhere in formula 1) A* awe? k are connected, in formula 2) k* and /£*, since Ζ Ix, %"'] is α Ζ \_x\-

module, and in formula (3) A* and //#, since Ζ is a Z Ixj-module.

It i s p o s s i b l e to find a number of other formulas connect ing A*, A , Κ*, Κ , //*, // and a l s o

Kunneth formulas for the d i rec t product Υ, χ Y2, s tar t ing with the complex Sy ® n Ζ L%J as a Zlxl-

module and the fact that Ζ [χ] i s one-dimens ional , a s is 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 cobor-

dism 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 funda-

mental cycle and the Cech operation.

We note that the homomorphisms at, introduced by the author on the ring Ω[/ represent "char-

acteristic 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? What are the Λ-modules £/* (BG)

and k* (BG), where A = Ωυ and Λ = Ζ [χ] ?

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 S2""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 de-

note 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, correspond-

ing to ρ - 1, with the relation Ψ"1 (t) = 0, where Ψ™ is the Adams operator.

We consider the ring k* (P) •= Hom* [ 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.

b) k2' (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).


d) There exists a natural generator u £ k2 (BG) such that every element of It' (BG) has the form

^ y"} U<lj and there is the relation (m^m)(u) = 0, or Ψ"1 (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 discus-

sion of the spectral sequence with term E2 = Η (BG, Ζ lyi) which converges to k (BG).

/ τη \We denote the expression πιΨ"1 (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 multi-

plication by y and has the form:

du = 0> dv=Fm(u).

The cell complex of k-theory S/C(BG) for G = Zm in the natural cellular subdivision has the form

Sic (BG) = Hom^r y](Sk> Ζ Lyi), while Sjc(BG) over Zlxi, Zlxi = H o m ^ r -,(Z lyi, Zlxi), is a complex

of free 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 iCPn]un+l/(n + 1) is the "MiscenkoU

series" (see Appendix 1). We denote the series ίπΨ (it) by Fmjj(u). Let

Λ = U*(P)= ΗοτααΌ(Ωυ,Ωυ),

and let S\j (BG) be the cell complex in U -theory which is a complex of Λ-modules, with Λ' 2 1 = Ω2'.

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 com-

plexes 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 multi-

plicative 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 52""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 C Zm and there is defined a homomorphism

U*(BZni)-^ U*(BZm). 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·


It is e a s y to show that for any (xi, . . . , xn) €. Zm χ . . . χ Z m

un (xu ...,χη)φ0 mod U. (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

by υ2π-ΐ ^- ^ 2 η · ι ( ^ ) t n e so-called "geometric bordism" a n ( l , . . . , 1). In the complex Sy(BG) de-

scribed in Theorem 1, the element i^2,1.1 is adjoint to vu""1 € Sy(BG), i.e., (i>2n.,, i u™"1) = 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 I

2) If Zj «fti^iji · · · 1 xhi) ^ 0, 2j an-k(i/ft4-l,j. · . · , i/n,j) ^ 0, then we have the equation:

3=1 j = i

ZJ an(Xij, • • • , Xhi-, yh+i,s, · · · , JJn.s) =s 0 mod Z?Zm,.

3', s

This follows in an obvious way from the fact that transformations 7\ : Μ -> 1/ and 7'2 : ,1/ -> ,W induce

(7\, Γ2): Λ/ χ Λ/π" , 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.

Lemma 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 (obviously);

u x2) = {XiX2)v3;

ft3 (#1, Z2, ^3) = (a-'l 2^3) Vs + ' · [CP1] VS.

910 S. P. NOVIKOV

Suppose given a group of quasicomplex transformations Z p : Mn -> 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

We consider the point (ΧΙΛ, • • • , Xkj, • • • > xnq) G Zp" up to a factor μ C Z p , μ φ 0. Thus, {χίΛ, • • • , xng)

G Pqn~l

m xhe group Sn χ Sq, where S^ is the group of permutations of k elements, acts on Pqn~ .

Definition. By the type of the action of the group Z p on Mn 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 pro-

jective space Pqn~1

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

Corollary. For p> η and for η = 2 ,3 , the set of types of actions of Ζ p on Mn is given in Pqa~l

by the set of equations:

Σ π, IT. • τ Λ — 0 -r- =iL· Π h 9 "ί η

where the xsj are the orders at the point Pj and o^iXij, • • • , xnj) i s 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 spec-

tral 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 alge-

braically 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, Zp) = E//A'^and 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 /·

,. . . ->- Hh> Ρ (Χ)-> Η"· Ρ+1 (Z) -+ Hh- P+1 (Y) -+ Hh· P+1 (X) ->-. . .

II) For X = K(Zp) the bigrading coincides with that of Cartan.


III) The cohomology A -module // (A, Z p) is Jjigraded, if in // (Α, Ζ p) 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' = Ext4' '^(//* (A), Z p)

for X Ε Sgr is arranged as follows:

d • Fs'h- β_>_ Fs+r' h' p i f r ~ 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 sub-

stitution.

Examples of bigradation (the simplest). Let A = K(Z) + EK(Z) and Υ = Κ(Ζ q). From the or-

dinary 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 ) = h^u , 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:

a) H**(X, Z p ) Α/Αβ(α) + Α/Αβ(ν), where u €. //°'°, ν € Ζ/0'1. Hence u* C E x t / ' 0 , ν* Ε Ext0 '1

and hoU G E x t 9 ' 9 ' 0 ; by dimensional considerations, dq (ν ) Ε Ext 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), u 6 H°'°, ν € Η 0 · 1 , then υ* € Ext0/ '0 and v* € E x t ° / ' \

Φ 0 for i = q.

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) (SN) of the same

order; moreover, nlp)(SN) = E x t ^ + . . . , where Ext1 '* = / * ® ZQP.

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 Ext2 pJ '1 '2 p 2 ( p ' l ) + 2p"2 ^ Q_ I n r e a l i t V ; t h e s e e l e m e r i t s i n

(/-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 argu-

ment for the existence of an invariant second grading in the subcategory Sgr C S ® zQp-

Received 10 APR 67

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