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THE METHODS OF ALGEBRAIC TOPOLOGY FROM THE VIEWPOINT OF COBORDISM THEORY
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Please note that terms and conditions apply. 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 sequence in cobordism theory, calculation of the ring of cohomology operations in this theory, and also a number of applications: to the problem of computing homotopy groups and the 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 [I7]; the most com- plete and systematic account together with the structure of the ring can be found in [I8]. 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
AcF. The ring over Qp, Λ ® ZQ p D Λρ, lies in A V 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 that H* (Mp, Zp) = 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 U υ υ A = AP = 2 τηρ*Αρ /τηρ**Αρ , where Λρ is an A -module. In this situation arises as usual a spectral sequence (Er, ar), where /(A, \)®zQP, E2 = Ext^(AP, 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* (Zp, Zp), and the algebraic spectral sequence (Er,dr) is associated with the "geometric" spectral sequence of Adams in the theory H*( , Zp). 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/ll/(Aj Λ) 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, Zp) 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 (Er , dr), if the sequence (Er, dr) is N trigraded by means of the Cartan grading, as is {Er, ar). Therefore the orders \n (S )\ 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 sequences A'-*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 0H.^4S->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). METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 831 2. For any short exact sequence 0^,44 B^C -> 0 and any Τ €S there are uniquely defined exact sequences T A)^Ri(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 +0 Y = 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 + γτ 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 h fn+h-l •••fn'· E Xn-^-Xn+h (direct) h fn· • ·fn+h-l'· Xn+k-*~E Xn (inverse) which allow us to define passage to the cofinal parts of spectra. For spectra X = (Xn, /„) ancj Υ = (Υ^ gJ we define Hom*(X, 7)= lim limHom*(Xn, 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 = XZ; of objects Xi CS lies, 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* (Zn+1, X), METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 833 <- Ύι where X GS C S, all Xt are equivalent to the object X or Ε Χ, 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 «~ Ύί where A; is Ε Χ, and therefore lies in S, and the ,4x-module Horn* (Χ, Έ.Χ;) is free. y By X-free objects for X €S we mean direct sums Σ^f£, where X; = £ 'A for arbitrary integers Ji- Finite direct sums belong to S. There are simple properties which give the possibility of constructing the Adams spectral se- quence by means of axioms 1-3 for the category S. For any object Τ €S and any X-itee object Ζ GS we have Horn* (Γ, Z) = Homl* (Χ* (Ζ), Χ* (Τ)). Let us give some definitions. 1) For an object Υ € S we understand by a filtration in the category an arbitrary sequence of morphisms h f, Y=Y.l+-Y0+-Yi^...*-Yi*-.... 2) The filtration will be called A^-free for X €S if Zj CS are Λ^-free objects such that there are short exact sequences o -»- Yi -t Yi-ig X 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 az: X* (Zi) ii- Χ* (Fi-i) -i X* (Zi-t) and 5 K δΓ: T.(Zt)-+T.(Yi) -m~Tt(Zi+i). 4) An A*-free filtration is called acyclic if (Cx, dx) is acyclic in the sense that HO(CX) = X* (Y) and Ηt(Cx) = 0 for i > 0. From the properties (axioms 1 and 2) of the category S and Lemma 1.1 we obtain the obvious Lemma 1.2. 1) Each filtration (Y "~ Yo *~ Υί *~ . . .) defines a spectral sequence (Er, dr) with term El = Bj, dl = δγ, associated with ΗοΓη*(Γ, Υ) in the sense that there are defined homomorphisms 0 qa: Hom*(7\ Y) - E ^* , q- Ker qiml -^ E^*, where the filtration (Ker qt) in T*(Y) = Hom*(^ Y) 'is defined by the images of compositions of filtration maps Γ* (Υί) -> Τ* (Υ). 2) If the filtration is X-free, the complex (Β γ, δγ) is precisely Hom*Ax(Cx, X* (T)) {with differ- ential 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 s s an integer re such that Hom (T, Xm) = Hom (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 l Υ -» ΧΧι factors into the composition Υ —^,/j Xn "*~ Zl E X, 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 (n (n> Obviously X* (Y0 ^) = KetX* (f0) and Yo 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: i t a) Ext - AX(X*(Xn),X*(T))=0, i> 0, t-i £ b) Hom^xCY* (JVJ, Ζ* (Γ)) = Hom (r, X) for t < /„, and /„ -» c» as re ^ ~. The so-called Adams spectral sequence (£r, Jr) 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 METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 835 where the Υ JY ι +l are sums of objects Xn, for numbers η which may be taken as large as we want, with the filtration acyclic through large gradings. For such a filtration,the corresponding spectral S l 5 sequence (Er, dr) has the term E2 ' = Ext^A » '(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, s s +r> + r 1 one can construct an Adams spectral sequence (Er dr), where dr: Εr ·' -» Εr ' ' and the groups _2J En are connected to Homm (T, Y) in the following way: there exist homomorphisms t—s=m i +m qf: Ker qi-i -*• ^ -j , i > 0, where is the natural homomorphism. The Adams spectral sequence is functorial in Τ and Υ. 1 Remark 1.1. The homomorphism g,: Ker q0 -» Ext '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 p q p r q + r +1 this spectral sequence, dT: Er ' -» Er ' ' , and the homomorphisms qi are such that where n q0:Rom (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 *-v XT- \r γ rr Z Qy 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. [ n i Let (//„ ) be the spectrum of spaces of maps //'„ = Q ' Xn = 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- METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 837 tains scalars with respect to multiplication X ® X-^X). Consider in the space H°n the subspace Hn C H°n ^Ω^λ",, which is the connected component of the element 1 GX°(P). We have a multiplication Η η Χ Ηη -> Η η \ 1 Ι Η°ηΧΗ°η^Η°η induced by the inclusion Hn CH°n. Let π {Κ, L) be the homotopy classes (ordinary, non-stable) of maps Κ -> L, and let Π"1(Χ) = lim n(K, //„). Obviously Π"1 (K) is a semigroup with respect to the previously introduced 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 Π0 (K) = X1 (K); therefore in the S-category Π0 (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). n An important example. Let X = Ρ = (S , *). Then the spectrum Ηη with multiplication //„ χ //„ n n -» Ηn is homotopic to the spectrum Hn (maps of degree +1 of S -» S with composition Ηη χ Hn -» //„). The /-functor of Atiyah is the image of K(L) -> Π0 (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', Π0 (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 1 λ^-homology manifold is meant a complex Κ such that X (K) = aKn_i(K) in the presence of some natural identification of σ Χ* (Κ) with σΧ*. (σ Κ); for example, if Κ is a smooth manifold, then σ(Κ) according to Atiyah L"J is the spectrum of the Thorn complex of the normal bundle in a sphere. In the presence of a functorial Thom isomorphism in X*-theoty for some class of manifolds, we obtain Poincare-Atiyah duality. Let X G S,, Υ 6 S, Γ € S. In V 1 we constructed the Adams spectral sequence with E2 term equal to Ext7* (X*(Y),X*(T)). 838 S. P. NOVIKOV The law of duality for Adams spectral sequences reads: The cohomology Adams spectral sequence (Er, dr) with term E2 = Ext^^ (Χ* (Υ), Χ* (Τ)) is 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. ls Ύι Proof. Since a stable spectrum X in the category So = S ® zQ equivalent to a sum Σ Ε Κ(Ζ) of Eilenberg-MacLane spectra for π - Ζ, and since for X' = K(Z) the ring A ® %Q ls trivial, it fol- s lows that all Ext 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 Zp. 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 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. 2n a) Let k = (kn), where il k2rl = 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. n b) Let kO = (kOn), where W k0sn = BO χ Ζ, and all k0n are (n-l)-connected. We have kO ^ l Sn 1 = (.k0j- h, where n koJ-n .} = BO χ Ζ, kO^ = W and the kO J ^ are (n-l)-connected. Here i is to be taken mod 8. l It is easy to show that in the category S <8> χΖ [ /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 0a((l), where a(rc + 1) > a(re) and a{n) -» °o for η -» ~, with Gn C Gn + 1 under the inclusion 0a C 0a(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: λ μ 2) MS0®z Ε> 3) MG ®zQP =Σ k 840 S. P. NOVIKOV where Η (Λϊ(ρ), Zp) = Α./βΑ +Αβ, A is the Steenrod algebra over Zp 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 λ4 corresponding to certain partitions ω (see [ *5, 17, 18, 26]). μ 4) Μ Spin ®z Facts (1) and (2) are known, and fact (4) is given in a recent result of Anderson-Brown-Peterson [10]. n c) G = T, where T = Gn C Un C02n is the maximal torus. This leads to MG, again a multiplica- tive spectrum since MTm + n = MTm (g> MTn. 0 Let us mention the structure of the cohomology M*(p )(/>), where Ρ = (S ,*) 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) 6S ® 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) an ® zQp 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 x sequences coincide, follows from the fact that MU ®zQp =-2 E kM(V), as indicated in §2, since k a sum MU 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). METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 841 Now let Υ satisfy the hypotheses of Lemma 3-2. We have Lemma 3.3. There exists an X-free acyclic resolution for X = M{p), MU: Υ <- Yo *- y,*~ . . . *~ Υ l ·- . . . , where the stable spectra Υ £ 6 Dp 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 m of the group H (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 Fo 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) in§l. From these lemmas follows Theorem 3·1· For any stable (m — l)-connected spectrum Υ € D C S, X = MU and any finite complex Τ € S of dimension n, the Adams spectral sequence (Er, dr) with term E2 = Exttx(/V* (Υ), Χ* (Τ)) exists and converges exactly to Horn* (T, Y); moreover Ext.£ (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 , M(p)(p), since The theorem is proved. 5 Note that for X = MU, M(p), stable spectra Υ and finite complexes T, all groups Ext '^ 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 s + r i+r fact that dr(Ef'')CE r · -'. Q.E.D. Corollary 3.1. ForX = MU, Υ = Ρ, Τ = P the groups Ext^' (Χ* (Ρ), Χ* (Ρ)) ®zQp = Q for are t < 2s(p—1) and for f ^ 0 mod2p — 2, and the differentials dr on the groups Er ® zQp 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 (Γ) r) p q r r + 1 they are finite in number: ϋ, , . . . , uj € E*·". Note that dr(E r· ) C £? + - «* . 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. r 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 , where Εr is a Λ^ -module with the finite r) ir) ir) number of generators u^,..., u\ , above. Consider dr(,Uj )= Σ k^uk , where λ^° G Λ. Let (r) (Γ) dim λ, . < /Vr for all k, j. Set Nr+1 = Max (/Vr, Nr). Then λ^, 6 ΛΛ By virtue of the Noeth- r +l r+1 erian property of the ring ANr t, the module Η (Er ® zAN r ,dr) is finitely generated, where ANr Γ ι r 1 Γ is generated by polynomial generators of dimension Nr < k < Nr + i and ANr + <8> χΛ + = Λ . Since N N r +i N r H(Er, dr) = Er+i = Η (Er ®zA *, dr) = H{Er ®zAN r ® A +i, dr) N I+1 Γ 1 if we set ΕΓ + ι = Η (ΕΓ ® Λ^Γ + , dr), then £r + 1 is a finitely generated A/vr + i-module, and Er + l METHODS OF ALGEBRAIC TOPOLOGY FROM CORDISM THEORY 843 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- l hilate the generator. From this and the lemma it follows that Ext '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 λω ; 0 MO = 2 Ε Κ(Ζ2), where ω = (o1; . . . , os) Σα; = λω, at φ 2 -l, or ω = 0. The Steenrod ring A ω is an algebra over the field Z2. 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,, . . . , as), (a[, . . . , α^), αω, ω' €• A and dim a = λω — λω'+ dim αω< ω' is the dimension of the matrix.. The ring GL (A) is, by definition, a graded ring. This describes the ω ring A more generally for all spectra of the form ΈΕ Κ(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 l 0* (K, L). Let Λ = 0* (Ρ) = Ω0 be the unoriented cobordism ring, 0\P) = il Q. 1. There is defined a multiplication operator X-+OX, xeE0*(K,L), oeA = 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 classes If; for all 0-bundles. As usual, the classes Wt define classes Wω for all ω = (α,, . . . , as) 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 Sq'» = <£ (IFJ, where ίτω € 0* (B0n). Under the homomorphism ϊ •]* : 0* (MOn)-+ 0" (BOn)-+ 0* fj]' ΛΡΑ°° j the element u = Λ* 1 goes into i* /* (it) = ii, ...«„, where u£ € 0 (RP°°) is the class Wl (£;), £. the canonical 0,-bundle w over RP°° , defined above, and Sq^ (Ui . . . un) = S (u^ .. . , un) ul . . . un, where SM is the symmetrized monomial Eu"1 ... u"s, s < n. 1 1 l There is defined the subset Map (A", MO,) C O ^) and a (non-additive) map y: O (X) -* H (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(*) = χέ +1 ; 2 c) the composition Sq ' oSq is a linear combination of the form SA^jSq", λω C Z2) 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) = OQ. 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 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, METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 845 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 εΩ0, ^(α)6ί1ο. It turns out that the following formula holds: Sq*.a= where a € Λ = U,Q. We also have a diagonal 01 δ( ω 2 so that 0 where Δ (α) = α (8) 1 = 1 <8> α, and ASq = 2 ϊ ' ® S^ ' Λ° <8> Ωο Λ 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), ; 2 Δ (α) = φ(/*2), where h2 = y(Wi) . 2 It turns out that d = 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. 20 21 The homomorphisms d* and Δ*: Ωο -» Ωο coincide with the homomorphisms of Rohlin [ ], [ ] and Wall [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; 0 2) Ext ·*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) Ext''^ + S = 0, for s jt 4k; 1 +4/t 4) Ext ' ^ (0* (MSO), 0* (P)) is isomorphic to Z2 + . . . +Z2, where the number of summands is A. equal to the number of partitions of k into positive summands (A,, . . . , ks), ΣΑ£ = k; 1 5) there exists an element h0 € Ext '^ associated with multiplication by 2 in Εχ, such that 0 1 1 t M 1 +1 !t Ext ' ., ->° Ext ' rj is an epimorphism, t = 4k, and Ext'· ' ^° Ext ' 'is 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)), 1 and H*(MSO, Z2), as was shown by the author t ?, 18] and by Wall [23], is 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]. sk+l In the study of Ext AU (U* (MSU)) all dimensions will be doubled, the groups E^ for 1 < i < 3 1 1 will be constructed in an identical fashion, but the element h0 € Ext ' will be replaced by an element 1 h € Ext '^, 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 2h calculated easily for [RP ] € Ωο and (ωι, ω2)=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 η U*(X) = 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, {) = (Ya , f-fa), where (Y a, fa) C i/* (A!) is the element having the form D α(-τχ), where τ χ € Κ (Ζ) is the stable tangent t/-bundle of X. \Da(-rx) = D((~rx)* (a)).\ As we know, the operation of the class α on ί'* (Κ) can be defined in another way: since V* (MU) = U* (BU) by virtue of the Thom isomorphism φ, we have φ (α) = α (Ξ U*(MU) = A . We consider the pair L = (K \J P, P) in the S-category; then i/* (K) = Horn* (P, MU ® L) by definition, where Ρ is the spectrum of a point. Every operation a = φ (a) defines a morphism φ(α):Μυ ^>MU and, of course, a morphism φ(α) 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 1 where a = φ' (a), a eU* (MU) and (Ya, fa) is the element of U* (X) equal to Όα(ξ), α € 6'* (Βί/), and ΓΜ is the stable tangent (/-bundle of M. T if ζ = X, 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 oi is also induced by some characteristic class and hence has the form a b = c. Whence follows the lemma. Remark 5.1. It is easy to show that the transformation a has the form φ~ια*φ, where φ : U*(BU) -» i/* (MU) and a*: U*(MU) -* U*(MU) is the transformation induced by a: MU ->MU. In the future we r\_, shall use the geometric meaning of the transformation a = φ' α*φ and hence we have given the defi- nition of ο in a geometric form. The transformation a induces a transformation α*: Ω;/ -> Ω(/ = V* (Ρ), where U\(P) = Ωί, = U'l(P). We shall also denote by a* the dual transformation U* (P) -> U* (P), U* (P) = A = Qy. We shall now indicate the set of operations needed, from which we can construct all the operations of the Steenrod ring A . 1. Multiplication operators. For any element α € U* (Ρ) = Λ there is defined the multiplication operator χ -> ax. Hence Λ C A . The corresponding transformation a : ί/*(βί/)->ί/*(βί/) has the form: where (Υα, —τγ ) represents the element Da €.U*(P) = £ly. 2. Chern classes and their corresponding cohomology operations. As Conner and Floyd remarked in L^M, if in the axioms for the ordinary Chern classes one replaces the fact that ci(^) for the ca- nonical f/j-bundle over CP is the homology class dual to CP 'l, by the fact that the "first Chern 2 )A N 1 class" σ1(ξ) is the canonical cobordism class ai €i/ (C/ ) which is dual, by Atiyah, to [CP ' ], then there arise classes σ^ζ) € U2l(X) with the following properties: 1. at = 0, { < 0; σ0 = 1; σ£ = 0, i > dimc ξ; 2. σ, (ξ+η)= 2 0}(1)σκ(ΐ\)\ 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 σω(| + η)= ^j Ρω^ξ) σω2(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: METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 849 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 σω(ο), represented by (Όσω(—τχ) μ ίwhere e is induced by X -> P\ are the characteristic classes of the tangent bundle. n Let X = CP and ωω = 2 ti'· · • U (the sum over all symmetrizations, ω = (klt . . . , ks)). Let λω be the number of summands in the symmetrized monomial ιιω, k = ΣΑ^. We have the simple n n k Lemma 5.5. If X = [CP ], then σ*ω(Χ) = λω [CP - ] and σ*(αδ)= 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. n Proof. Since for X = [CP ] we have that τχ + 1 = (π + 1) ξ, where ξ is the canonical ί/,-bundle, 2 n 4 the Wu generators for τχ are u = i, = . . . = tn+ 1 = DCP"·' €i/ (Z). Therefore σω* [CP ] = λωϋ , where k = dim ω. We note that by virtue of the structure of the intersection ring 6f*(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 1 CP? x ... x CP™- * BUn-^MUn and homomorphisms i': U 2 We note that U* (CP™ χ ... χ CP™ ) has generators u{ € U (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,, . . . , un), 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: 2 k + 1 1. If a € Map (X, MUJ CU (X), then S(k)a = a and Sw(a)= 0 if ω φ (k). 2. So (αβ) = 2 &>. (α) ^ω2 (β) for all α, β G U*. ω=(α>ΐ, ω2) u) (l ( λ then 3. Ifk 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 δω. Proof. Let X = BU\ = CP°°. Since MUl = CP™, it is sufficient to prove property 1 for the element 2 u GlJ (CP™) equal to σι (ξ) for the canonical f/j-bundle ξ. By definition, we have: it = ]*φ(1) 2 c + 1 £ U (CP™) andSa)(Ii) = /*Sa,0(l) = /*u' (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) asn^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) = Σλωδω(κ) and then use property 3. Whence it will follow for large η that Ξω oS^ = Σλω5ω. 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[/. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 851 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, dim*; = — 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 (5ω) = Σ ^ω, ® 5ω,· (ωι, ω2)=ω Since MU is a multiplicative spectrum MU ® MU ^MU, the ring /ί has a "diagonal" υ υ Δ: ^->^ ®Λ4 , where Δ(5ω)= 2 ^ω, ® Sa>, and «α Θ i = α Θ zi> = %(o ® 6) for « € fly = Λ. The Kiinneth ω=(ω!, ω2) formula for Ky, K2 GO {complexes without torsion! 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 class, and γ(χ) € Map (A', ML',) C U2 (A), for χ € U2(X). We consider important examples of cohomology operations related to the class 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(10) 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 it: Yt -> X, i2:Y2 -> X are submanifolds which realize the classes Dcx {ξ), —Dcl (ξ) e Ηn_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 ** (^ + ·;·+.ξι + ίι +y+.ii) = 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. ust as was tne 2. The classes and operations X(kli k2)· J case for the operations &(klt k2) and 2 classes γ{ο% ') y(— σ,) , 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'is equivalent to the sum k times We set {here kl + k2 = k\ rr n which functorially introduces a £/-structure into the bundle τ'(klt kz) such that '(ki, fc2) = where r is the realification of a complex bundle. n Ρίξ+k) has the induced t/-structure p* r(CP ) + r\ki> ki). We denote the result by Ρ^·^\ξ (k n n + k). We denote the pair (P >- ^ (ξ + k), p) € U*{CP ) by 0χ(Αι, ki), where X(kl, k2) € U* (CP ). 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. e and There arise classes X(klr k2) ^* (BU), operations φχα^, k2) = ^(klr k2), 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 t1?]). It is easy to establish the following equations: 2 a) A(ki kl)°d = 0(in particular, d = 0, Δ<9 = 0); 1 U b) ΔΨ= 1, [d, χ] = 2, Xd = Xlod, where ^ = [CP ] € Λ C A ; 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). METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 853 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 ct(.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, k if α£ = S/,. and b^ = [CP ], 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 U +1 theory U* into a sum of theories nfU*, where Σπ; = 1, 77, € A , with n0 = 1 -ΨΔ and 77, = ψ>Δ' -Ψ>' Δ;+ι. Later on we shall meet other projectors of this same sort. 3. We consider still another important example of a cohomology operation in (7*-theory, connected with the following question: Let ζ, η be (/-bundles. How does one compute the class σ^ζ ® η)? We have Lemma 5.7. a) For any Un -bundle ζ there is a cohomology operation yn., € A such that σλ( = Yn-ι (ση(Ό)> where λ_, = Σ( — 1)'Λ' and the Λ1 are the exterior powers. 2 2 b) If ul, . . . , un G U (X) are elements in the subset Imy = Map (X, Ml)\) C U (X), then we have the equation Yn_l(lii ... Un) = Υι(«1·γΐ(Μ2·γι(...·νι(κη_ι·ΙΙη)) ···). where y, is such that for a pair of Ui-bundles ζ, η we have the formula The proof of this lemma follows from the definition of the operation yt. Let X = CP°° χ CP°° and let ζ, η be the canonical ί/,-bundles over the factors. Since νσ^ξ ® η) = c^t; (g) 77) = c^tj) + £^(77) and CTt € Map (,Υ, Λ/ί/χ) it is possible to calculate the class σι (ξ ® 77) completely as a function of at (ξ) and σ, (77). Namely: Since the bundle λ, (ζ + η) lies in a natural way in K" (MU2) and λ! (ζ + η) = ζ ® η — ζ — η + 1, the difference -σ, (f) -σ, (77) -KJ-J (£ ® τ;) + 1 has the form γ^, where u G U* (MU2) is the fundamental class u = Yjii = 2 a;j,Ai,j)(«i"2), Μ = 854 S. P. NOVIKOV where ut = 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 ση (^) for a £/n-bundle f. The lemma is proved. 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 yt' = 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 = at (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 €Imyt = 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 = METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 855 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 2n 1 "Riemann-Roch" transformation λί"*: U ->K which is defined by the element λί"" € K(MUn), and let λ = (λί;°), η - oo. Let Φ(π) = σ/ί" °λ^), and Φ = (Φ(η)). The operation Φ obviously has the 7 property that Δ Φ = Φ ® Φ since ah and A.t are multiplicative, and if the element ^C/iiA ) 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,Ϊ6Λ,Λ = U* (Ρ), Φ*: Λ - Λ, where 25 "^ >' Hence, the operation Φ associated with the Chern character ah defines a projection operator, which selects in the theory U* k a = "y. So,, a(u) = ; for ah = 3^! 5ω , a(u) = u (1 + u ). ^ ί — u ί-Λ « o>=(ft h) For our subsequent purposes the following operations will be important: U 1) The analogues of Adams operations Ψ^ £ A 2ί 2) Ψυ • χ = tex • ψ£. r«e χ e Λ" — Ω"; fc Μ® ...θα 2 3) Ψ[/(Μ) = — (A times), where u € U (CP ) is the canonical element and k 2 composition in Map(,¥, Λ/ί/χ) C U (X). Lemma 5.8. a) The series xH.,(u)hasIk the form - υ 1 = —f(u,f(u, ...J x ' k / b) Ψ* * (x) = kix, χ e A~zi == υ-Μ(Ρ) = 856 S. P. NOVIKOV j\ XI) k ψ I __ m kl __ xu I m k . a> U U ~ U ~ UΎ U ' e) for a prime p, all λ;, λ; G A'21, such that π 2n are integral for i < p. Hence the element ρ Ψ£ (ut ... un), where ux . . . un € U (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 Xi = [cpq, JI(U) = (ι + 2 *,-»*)«. >1 where the λ; € Λ ® Q are polynomials in xt with rational coefficients, dim λ; = — 1i. It is easy to 2 show that π *(λ;) = 0, since π = 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. y Let y € Λ and Δχ C Λ ® (? 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) Ay*(y)= l. U LetOy =yAy, Ψχ = 1-Φχ € A <8> (?. It is easy to see that Φ^ = Oy, Ψχ =Ψy,andΦyoΨy = 0. l l l +1 Moreover, the collection of projectors π£ = J A y — y is such that πρ^ = 0, / Φ k, and it decomposes the theory U* ® Q into a sum of identical theories. ; ; Let y] eA = Ω^ be a system of polynomial generators, and Φ£ = yiAy .. We note that Φ* (y;) ψ = 0 for / < i. Let ^ = yk for A < / and y4 = (1-0ί)*7Α = ί* (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. +1 +l :Λ Λ The projectors nf = y/&' -yi' Ay.' clearly are such that 77*/ ^ carries monomials of the form y .'γ\ , .. ., γ ι , j > 0, into themselves for t,, .. . , is Φ i, and all other monomials into zero. This means that and Kerjt/= U In particular, 1— Σττ;· and ττ;+ 1 = y^jAy.. Hence = ?r;-(y^) for all % e ί/ , and all theories / iTj (U ® ζ*) are isomorphic. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 857 The projector na = 1 —y,Ay[ 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* 2 2k Examples of operators Ay: if dimy = 2k, i.e., y G Ω[/ = U' (P), and ff*ft>y = - λ ^ 0, then we set y = 2J For the generators Ji € we have | λ| = 1 for i ^ p' — l and | λ| = ρ for i = p' — 1 for any prime p. Hence q is v{ = Σ yi ~ a,...,i and q-1 r ; It is easy to see that for i + 1 Φ ρ' for given p, Ay . € /I ® Z*?p> i° i + 1 = p and ρ > 2, Δ € /I ; 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, since φ and ί: = 1 -Jik^Yi,' ^y,, commutes with yn j < ih. We consider the operator ΦΛ.Δχ ΦΛ = 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/ "^. . ^. a ; + 1 + 1 ties making 1 — yih a t^lc.i = ^k-ι multiplicative projector, and Φ,/ ,= γ'. Δ — y' Λ' forms a ft*i • ^-^'^'^h'lk-i A; "i completThuse ,system of orthogonal projectorsΦ. = Φ (1 -y -y Λ Λ I h and = Φ4-ι, while lfc., /£i here Φλ = 1 -yiA.Ay .^. If where ^, or: 858 S. P. NOVIKOV 1 1 φ™,, = φ™ φΗφ™ ,..., φ}" = Φ^Φ,Φ^ ..., ΦΙΜ = φ, * [ft.] The projector Φ is obviously such that ys, s + 1 = P> for S s=; ift, υ b) φΐ*] e= ^4 ®ζ Qp. The collection of Φ' 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 —φΜ) = φ defines a projector Φ € /I ® zvp which is multiplicative and such that: a) O*(ys) = b) Φ2 = Φ, U to a c) the theory U* <8> %QP splits into a sum of identical theories of the form Φ(ί/* ® z@P) P shift of grading (suspension). We note that the elements γs = Φ*(γ!.) for s = p' — l have the property that all σ£ (ys)^0 modp for all ω, dim ω = 2s. The cohomology theory Φ(ί/* ® zQp) is given by a spectrum Lemma 5.9. a) There exists a multiplicative projector Φ € A ® z(?p sucA ίΛαί ί/ie cohomology s theory Φ(ί/ ® zQp) ' 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 n 2 2 1 2n X = Ρ (a point), A: = CP , X = ffP ", X = RP "· , Ζ = MSU, X = S -VZp, X= BG, G = Zp. 7 1. Let A = 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 U hx gHom^ y(A , Α), χ G A, and hx is such that hx{\) = x. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 859 In particular, for a = S^ we have α* = σ*ω , and for a = d, Δ we have a* = d* or Δ*, the known homomorphisms of the ring Ω[/. These remarks are essential for computing ExtAu( , 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 δω given in Lemma 5.6. l 2n + l k 2rl+ in) 2n + i { n) 3. X k»> = S /Zk =(E S VZk)eS, X2 = RP . U*(X) has two generators u d U\X k ), 2n+1 υ G U (X k), satisfying the relations: a) Ξω(ιι) = 0, ω £ (q), b) S(q)(u) = 0, q>n, c) (ΑΨ^Χ») = 0, u € Map atj Mf/,), d) Su(v) = 0, ω > 0. 2 These results follow from [7] for K* (BG), G = Zp) and the σ, K° ^ t/ and the ring U*(BG). 4. For X = #P2n, SG, the module U* (X) is described as follows: a) U*(RP2n) = U*(RP2n + l)/v. n) b) U*(BZk) = lim [£/*(^ )]. 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 (yaj, / 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 υ φΙ(γσι)€Α . 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->οΌη U*(BU). Let ζ be an SU-bundle. It is easy to see that we have the equation 2 1 where cl (η) is the basic element of H (CP ). 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 U h h 4 u k kl (A A ΘΛ Ζ) = #< - > (MU, Z), (A d ®A Z) = B (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 -modules e 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 2 2 for i > 1. We set rf(u£) = du^ and d(vt) = Au^. Since d = 0 and Ad = 0, d = 0. It follows from the theorem above that C is an acyclic resolution of the module U*(MSU) = H0(C). U We now consider the complex Horn*Au(C> Λ), where Λ = U*(P). Since Horn*AU(A , A) = Ων, we obtain the complex d* d* d* HomJXC, Λ) = where d* = d* + Δ*:Ω[/^Ωί/ + ily. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 861 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 Ών. ι c) If h £ Ext^'li (U* (MSU), Λ) = Z2 is the nonzero element, then the homomorphism α -> ha: Ext '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 1 of (c), we note that h = i^ d* OJ, where xt = [CP ] € 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. 4 1 2 We consider the element Κ = 9x1 - 8x2 € Ω ^, where xx = [CP ], x2 = [CP ]. 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). 3 It follows from dimensional considerations that d2(K) = 0 and d^(K) = A . 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 q q Lemma 7.2. // a. € Εξ· for ρ > 3, and d3 (a) = 0, then a = di (β). Hence £/· = 0 for ρ > 3, and V = F 3 3 Proof. Let ί/3 (α) = 0; since a = Λ β from Lemma 7.1, d3(a) = rf3 (Λ β) = 0. Hence d3 (β) = 0 P P + 1 since multiplication by h :£3 " -» £3 · is a monomorphism for ρ > 0. This means that α = άι(Κβ). Since P Σ £3 '* = 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^ . 0 is a Since E*J = £ ^* + Eli* + Ell* associated with Qsu = π, (MSU), and E^* = hE J\ 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 [18] by other methods, and(b)in [12L Corollary 7.2. a) 7*Ae image ofΩςυ ^n Ω;/ ls 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 2h 12h+2 d3: (Ker d* f] Ker Δ*)* = Es -+ Εξ = 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:. 1 2 1 2 2 1 3 3 where Ext ' ^ = Z?^' '' = ^J' * 3 ^^J = Kerrf 3,h' d} is well-defined since h is a monomorphism on 4 3 2 5 4 Ext^" , and the image lmh' d, = Ker d3 = Ω ,^ C Ext^" . 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]) l since H2k_2 (W, J We note now that the groups H*(W, d*) ace computed in [ 3]: namely, Hik{W) = Hski4(W) = Z2 + . . .+Z2 (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) Ext1',8*46 = Κ Ext1·'**2. where K € Ext0·?,^* (MSU), A). k + 3 h 2 b) d3 (Ext°'j) = 0 for i ^ 8k+ 4, and d3 (Ext°''u " ) = Ext -'(J + is defined by the condition 3 d3(K) = A . Proof. Suppose both parts of the lemma proved for k < k0 — 1. We show that d3(Ext°'y °) = 0. In 3 fact, by the induction hypothesis on the groups Ext '?, the differential d3 is a monomorphism. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 863 Hence Ext0·'*0 ^ 0. A u 8 0 0 8 0 0 8 0 We now consider d,(K Ext"· ^ ) = /i'Ext ' ^ . We see that d3(K Ext ' ^ ) is an epimorphism onto 3>8 c + 6 Ext3,eJc0 + 6_ Whence parts (a) and (b) of the lemma follow; on the group Ext ' the differential is trivial, and on the group Ext3>8fe + 2 D Ket d, = 0. The lemma is proved. 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^^/ΤΟΓ 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); 8 let TT2I = /<;. Now consider the numbers ch(cKti . . . cKik)A(X) [X] for Χ €Ω ^/ΤΟΓ C fl'y. These num- 8 1 l bers are different from zero mod 2 if and only if hX Φ 0 in Ω ^ . Hence the condition d3(KX) = h X^ 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 ch(c/ *^8. /c-theory in the category of complexes without torsion n Here we shall consider the cohomology theory k*, defined by the spectrum k = (kn), where i(kn) 2η = 0, i < n, and Ω 1ί2η = 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. 2 There is defined the "Bott operator" χ :k2n -> k2n_2 by virtue of the Bott periodicity £t k2n = 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 n 2 2n by the integer k (see [ J concerning the operation of Ψ onK° (S ) = 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 nX k s that (Α"Ψ ) can be considered as a map k2n -> k2n for which (k V )if: T2n+2j(k2n) -» "i n + 2 7(^2n ) * multi- plication by kn + >. :== η Λ Let αη ^λ^ (Α; Ψ ), where the A^ are integers, be an unstable cohomology operation ft and a^* multiplication by ^j λ& kn+K h Definition 8.1. The sequence a = (an) will be called a stable operation if for any / there is a number η such that for all Ν > η the number dy = ^jkh kN^ is independent of Ν. h Definition 8.2. If the stable operation a has a zero of order q in the sense that a* ; = 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 /1ψ , 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- k odicity x. This ring we denote by B 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 k under the conditions: 2 μΛ1>Αη+ί = 0, } < i. h METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 865 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 ι) which we denote by yt = Λ^δ,. It is uniquely defined by the properties that y, * = 0, Υι* = 2, and in o To Ti T2 0 1 0 0 . . . 2 1 2 0 . . . 4 1 0 24 ... 1 By definition, δ; = χ' 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 βψ 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. q) l q Lemma 8.1. The greatest common divisor of the integers γii = (x 8i)l ^ for all i > 0, for a fixed integer q, coincides precisely with the greatest common divisor of the numbers k (kq — 1) for all n + k. There exist operations a^jn G Β „, such that a^,,* = k > 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, n x k l and the determinant of the transition from the k ( \! — 1) to the x 8t is equal to 1. In fact, the process described above for constructing {χι$ϊ) is the process of reduction of the set of transformations n k k CV — 1) to the set y; of "triangular type" on Ζ [χ] = k (Ρ). More exactly: let re be sufficiently ( ( large that y . ^ = 0 for / < i and y. ^ = y^ for i < f(n), where /(re) -> °» as re -> °° and y. n = Σλ^.^^Ψ^). Under the condition Σλ^η?Α" = 0, one can write all these operations in the form Έ.μ^η^kn(Vk — 1) and then apply to the set /ε"(Ψ*— 1) the process of reduction to "triangular form" described above for l Π constructing the operations {x 8i) up to high dimensions. We assert that the passage from !Α (Ψ — 1)! to \γιιη\ is invertible. Indeed, any operation of the form Σλ^.Α:™Ψ has the form ΐιίγί>η + bl, where 1 2 •b[V = b[ *? = 0. Hence the operation it has the form bl = μ2γ2,η + b2, where b2V •- b$ = b[ * = 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 l coefficients μ£ stabilize, while an = 1i^ix8i + 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 l k the basis (x8j) = (γέ) it coincides with the greatest common divisor for the basis (k"C¥ —l)) = bk, where b^'* = kn (k' — 1). We note that the operations (knXVk) ate nonstable, but, by virtue of what has n + been said, there exist operations o^.) n such that a^J^* = k ' for / < /(re), where /(re) -> oo as η -» <Ό. These operations are obtained by the transformation from (x1^) to (λ"Ψ ) inverse to that described above. The lemma is proved. ί Remark 8.1. The same operations Ψ in k* ® Q are obtained as formal sums of the form Σμίχ δ£ 4 = Ψ , 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 ) A x k x k s = ((v. + 1) -1)' and Ψ(χγ) = H (x) V (y). The elements of k*(MUn) have the form Σλ;> sx ds, 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 k pair al € B , a2 € B^. of operations which have zeros of order qlt q2 respectively, we introduce the Qi Ql Ql +92 operations x' a1 = b1 and x' a2 = b2 and the composition b1 ob2 in Αψ . We shall show that x bl b liesinB^ ifx^b.eB^. Let a = ^ λ^knWh and a = Σ μ^^ΨΚ We consider o 2 ln 2n )( h h q2 q2X k x k 12 iusing k x V = H x' }. We shall assume that re is very large, re -> oo, qi and q2 are fixed. We set m = η — q2. Then 2ΑΛ METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 867 m) η) η) q where A^ = λ^ and μΙ = k μ^ • Clearly, asm» we have a composition of operations in Βψ which lies in B^. . The lemma is proved. 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 ρ k η ρ η ρ ρ by (ρ"Ψ ), with ρ prime, and by χ e B w , where ίρ Ψ )χ = ρχ(ρ Ψ ) and yp = (ρ"Ψ ) are polynomial k generators. Thus, a (topological) basis here is x P(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 l is equal to ip"(p —l)!p, where η is large, ρ is prime, and \ \p means the greatest common divisor of the sequence of integers. Proof. We construct a Βψ -free resolution of the module k* (P). Let η be large. Then the module n n k*(P) is given by the relations (yp—p )t = 0. We choose generators κρ = (yp—p ) 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 Cl; respectively. We consider the complex Honijt (C, k*(P)). 21 {p) Let ht € Horn k (Co, k*(P)) be^lements such that hL{u) = x\t), and ht C Hom^C,, k*(P)) be P B ( such that hl \up) ^x\t) and 0 = hi P\up^) for p'^p. Obviously, we have {d*hu llp) = (hi, Kpll) = χρχί (t) Hence, d*hi= ^\Ρη(ρτ—l)^i · Thus, is a 1-cocycle for the operator d*. 0»"(ρ*ΐ)0»(ρΐ)>> , d{hi) Since Hom*( , k*(P)) is a free abelian group, the element . —: , is the unique element l \P \P 1)/P 1 21 of finite order equal to dt in the group Ext ^ (k*(P), k*(P)), dt = ίρ^ρ'-Ι)!,,, η -^ «.. The lemma is proved. Note that the computation of ExtA (k*(P), k*(P)) presents no difficulties, since the module ψ k*(P) has a βψ-free resolution which coincides with the complex for the polynomial algebra Z[y2, . . . , yp, . . .], as long as the operator χ acts freely on k*(P), B^, . We have l l n l Theorem 8.1. The groups Ext '*(k* (P), k* (P)) are cyclic groups of order dt = \p (p— l)ip , 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) = Ext^ (A (P), k(P)®Q) = 0 for s > 0. Hence the groups Extjj'/,21 are all torsion. We consider the resolution Ψ I d π h £ .k β t \ ί ...->• 2iAw,i->-A where o?(u;) = Sj(u), and uiy u are free generators of Ct and Co. We consider the nonacyclic complex l where d(Vj) = x 8i(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 J In fact, if λ;· 6 Hom^^ (C, k* (P)), where hj{v) = x (i), then hi Thus, if 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 l \'^\i. Since the elements (x St) 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 of ψ 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 essentially 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 METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 869 a complete set of congruence relations on Chern numbers in Ωμ. More precisely: if LA j 6 il/j is an indivisible element, then there exists ξ € K(X) such that ch ξ Τ (Χ) [Χ] = 1. By virtue of the 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 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* iv\ . Λ —*• U (Λ-%) —>U \Λ.) —^(7 \A-l)—*• " I)) vhi ν I)) 1 0—» Horn* ft —> Horn* Λ —» Horn* h —>· Ext '! (k* (MU), ft*(X2)). "IB· "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* (X2)) 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 Ext1'! (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 s l Let kO be the theory defined by the spectrum kO such that Q, ' kOn = BO χ Ζ (see §3). The coho- ν mology ring of a point kO (P) = AQ is described as follows: generators 1 € Λ° h € ΛΟι 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 nX k their combinations α = (on), an = Σλ^" (Jt 9 ), 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 -» βψ, 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): y (kO* (P), kO* (P)) -y Ext^Io k0ψ "ψ and a Hopf invariant qx: Ker? 0 — ExtK0 (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 Exti#(kO*(P), W(P))=Zdh, dh= {pn(p^ - i))p and Exts^o (Λο,Αο) = Z for s = 8A + 1, 8k + 2. ψ 2 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)->- Ext^ (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. 0 1 3 There is defined an element h € Ext ' n (AQ, AQ) such that 2h = 0, h = 0, while multiplication nku by Λ ν 0,8h+l h 0,8ft+2 l,s l,s+l fe 0 ExtB o - ^ is a monomorphism for s = 8k, 8k + 1. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 871 The images of the homomorphisms 0,· and are easy to study: namely, q0 is an epimorphism (see L^J), and the image Im §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). The tln S Αβ contains the factor-ring A /] (B), where J (B) consists of all operations which vanish on the category B. We now define "the generalized Hopf invariant:" let g: K^KZ be a morphism in Β such that the object CKi [j K2 ( = 0 + j K, in the notation of §1, i.e., the sum with respect to the inclusions K1 — and Kl ^> K2) also lies in B. We have an exact sequence · C υ κ2) If the homomorphism g = q0 (g): X (K2) -· {EK^ is trivial, then we have Ο-νΓ(ί,)^ Χ* (CKi U K2) -*- X' (K2) -*• 0» g where Χ (Κι), Χ (CKl (J K2) are modules, and our short exact sequence determines a unique element g ?i(g)eExtA* (Χ*(Ki),X*(K2)). We thus obtain a mapping (Β) q± : Ker ςτ0 ->- Extif (Χ* ί^), X* (K2)), 7 ( B where qa:Hom* (Klt K2) -. Horn* Α" (Λ * (K2), A:*(K1)), K,, K2 € β and g 6 Kerq 0 \ 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 B) images ql (Ker qo )? If Α β ε .4 β is an arbitrary subring, then there is defined the usual homomorphism: 872 S. P. NOVIKOV i: EXU* (X* (K2), X* {Kx) )-^ Ext"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- B) 1 variant qi (g) € Ext X* (!*(&,), X*(K2)) is defined, and is equal to zero if and only if X* (L) Β = X (Xj) + X (K2) (as groups). Of course, examples 1 and 2 are uninteresting. We go on now to the examples which interest us. 3. Let Β = D (complexes with no torsion) and X* = H* ( , Zp). In this case A% = Α/(βΑ + Αβ), where β is the Bokstein hombmorphism and A is the Steenrod algebra (over Zp). There is a canonical isomorphism ExtA {Ζρ,Ζρ) — ExtΑ/$Α+Α$(Ζρ,Ζρ) for t > 1, where Zp = 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 . 1 The groups Ext · {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 ( Y) where P* (X) = Horn* (X, P), defined on elements such that q 0 • } = 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. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 873 Examples. 1. Let Χ , Υ be arbitrary cohomology theories. An arbitrary element α € Υ (X) determines a transformation of theories a: X* -> Y'. 2. If the theory X* is such that X1 (P) = 0 for i > 0 and Χ" (Ρ) = π, then there arises an 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 ® Zp). In the cases of interest to us, π = Ζ and rr<8)Zp = Zp. 3. The Riemann-Roch functor. Let X = U and 7 = A: ; we consider the Atiyah-Hirzebruch- η) Grothendieck element λ[ € K°(MUn). It defines a map λ_ι: U*-+K' (/ι) (π) 2n (π) and λ : i/* -. A*, where λ = (λ ), λ Ε k (MUn), is the element (uniquely defined) such that %"λ = 16 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 and 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,). q ar By definition we have: Η* (Hom^ y (aCx, a,V)) = R*GN(M), where R* = ^ R »d GN = Β 4 Hom^V ( , a.N)°a. is the composite functor, RqG is the g-th right derived functor. There is defined a Β 874 S. P. NOVIKOV natural homomorphism rq: Ext^x {M, N) -> R"GN (M), r = 2 rq, AB q and homomorphisms q q Pi': R GN (M) -> H * (Hornby (CY, aiV)) AB y(aAf( , aTV), AB where Ker /82* = 0. We have the composite map 1 a= (tir^W. E1(a)-*Ext -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). q In case (a) (a is an exact functor) there arises a spectral sequence (Er, dT), where Εζ' p — 2 R Gq,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 ( , Zp), the Riemann-Roch functor λ : U — k and the augmentations vp : U -> H*( , Zp). We have Lemma 9.1. a) The functors λ : U -> k and vp:U ->//(, Zp) are algebraic in the sub- category D; b) The functors λ and up are exact in this category. q ? c) The functor λ is such that R GN (M) = Ext υ {Μ, Ν), where Μ = U* (K2), Ν = U*(KX), Μ, Ν € D, GN ^i ( , λ/ν)°λ, λΝ = k*(Kj. q d) The functor vp is such that R G^(M) - Ext^ ,», ^»(i/pM, i'pN). METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 875 Proof. The category of A -modules corresponding to the category D is the category of Λ-free modules, where Λ = U* (P) » Sly. On the cohomology of a point Λ the functor λ is such that Λ-Ζ [xl and \(y) = T(y)x\ where y Ε Ω|/ = V'2i (Ρ) and Τ is the Todd genus. From the group point of view we have \M = Μ &Λ Ζ [χ], where Μ is Λ-free· There follows the q exactness of the functor λ and R \ = 0, q > 0. For vp we have vpll (P) = Zp, 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), ΐε 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. υ Now let α = vp : V* -> H* ( , Zp). In §5 we constructed a projector Φ 6 Α ® 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 k P (vpx) = νΡΦ5ωΦ (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 n l of order equal to the greatest common divisor of the integers \k (k — 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 asserted Ψ orders, Corollary 9.3 follows from Corollary 9·1· We shall indicate a simple fact about the connection between the Hopf invariants in different cohomology theories Χ , Υ in the presence of an algebraic transformation a: X -. Υ with respect to the rings Ag, Α β in the subcategory Β C S. Lemma 9.2. We have the equality _(B) ~ _(B) qiY = a-qix on Ker ?o^\ the group ^^'(Ker qi^) being contained in Ei(a), the domain of definition of the homo- 1 morphism a. = (β*' · β* · rr). The proof of this lemma follows immediately from the fact that by construction of the generalized Hopf invariants qlx and q^Y we can compute both quantities qfy (a) and qfy (a) for any ( a€ Ker ^'CKer 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 -. Υ AB 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 Λβ Ker gay » then the element γ is not realized as the Hopf invariant of any element of Horn* (K1, K2). 1 §10. Computation of Ext 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 U where Co = A (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) METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 877 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. 1 Now let i be odd. We consider the element xj, where x1 = ICP } € Ω|/. Since σ*{χχ) = ±2, all χ θω( {) = 0 mod 2, ω > 0, from the properties of the homomorphisms σ*ω described in §5. Hence the cokernel Coker d* always contains an element of order 2. Since the homomorphism Ext^'y"1"2 1 n i -» Ext '^ is monomorphic and \k {k — \)\]c = 2, i'= 1 (mod 2), we have Extl'"+2 (C7* (P), J7* (P)) = Extl'^+2 (ft* (P), ft* (P)) = Z, for 4i + 2 =2t, i= l(mod 2). Thus, we have proved 2 Theorem 10.1. The groups Ext'' 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 21 multiple Ayi; λ φ 0, represents an almost-parallelizable manifold M , 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/ -· S2' a projection of degree ±1. From the requirement of the integrality of the Todd genus and the 21 fact that (ch κ1, Μ ) = 1, it follows easily that all οωγι for all ω are divisible by the denominator of an a the number (β^/α;· 2l), where a2s+i = 1 d 2s = 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 Ext1'2' for I = 2s and has index 2 in Ext1'2,1 for / = 2s + 1. A i,8fe 1,2 3 d3(uh) = h ExW (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 , //* ( , Zp), 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) Erti$ (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 1 lf d) the homomorphism Exthko(Ao, Λο) — Ext^%(A, Λ) = Ext1;** , [ ^ i0 (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'. n Consider the groups πη + *k-i (S )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 t quh: nn+ik-i(S»)^ExtA^(k (P), 1T n has index 2 for k = 2l + 1 and index 1 for k = 2l. Moreover, the image q1>k( n + 4k-i(S )) coincides with the image quk • Ιπ^-ι (SO). METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 879 2. We now consider the Riemann-Roch functor λ : U' -> k and the corresponding homoraorphism λ : Extx'J. (Λ, Λ) -· Ext1'! • Since λ is a monomorphism, we get from item 1 on complexification the following conclusion: n 1 The Hopf invariant ql : nn + ik-i (S ) -> Ext '^ is always defined, and its image Im qly coincides 1 4 with q^f, (J π2ι(.ι (SO)); it coincides with Ext ' ,, (Λ, Λ) for k = 1, k ~ il and has index 2 in the group Ext1·8,^4 for /> 1. Later we shall study Ext1>e* + 2 and Ext1>8*+6. Au Au 3. We now consider the functor vp : U* - H* ( , Zp) and the corresponding Hopf invariant n Jtn+i_i (S ), η -> oo, i > 1 Γ ^β(Ζρ, Zp) = Ext^(Zp, Zp). 1 1 Since Ext ' (2D, ZD) = Ext/i(ZD ZD) for i > 1, this becomes the usual Hopf invariant. Since Α/ βΑ + Αβ μ ^ HP r the homomorphism qw: /jt8ft-i(^O)-^Ext^ (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 /^4i.,(S0). For example, let p = 2, and let 2 hi € Ext^' (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 2 Conclusion. For i > 4 the elements hi € Ext^' (Z2, Z2) do not belong to the image of the homo- morphism 1 2i v2: Ext ;* (A,A)^ The case ρ > 2 is considered analogously. In fact, we have the purely algebraic Theorem 10.4. The image of the homomorphism vp: 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 t t δ: Ext Jj(U'(P),U (P))^Ext'A*u(U'(MSU),U'(P)). We assume that Κ 6 Ext0^ (U* (MSU), Λ), γ Ε Ext0^ ({/* (MSU), Λ), and h €. Ext1·^ ({/* (MSi/), Λ) are 3 elements such that άζ(Κ) = A , 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 € m establish that all homomorphisms σ*ω(χί· Κ • y ) ate divisible by 2. It is easy to verify that σ%(χι), m σω(Κ) and ow(y ) are divisible by 2. The general result follows from the Leibnitz formula The lemma is proved. As was shown in §6, in the Adams spectral sequence for U (MSU) we have: m 4 m a) d3 (hKy ) = h y ^ 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 1 Theorem 10.5· a) The groups Ext '^. (Λ, Λ) = Z2 are cycles for all Adams differentials and belong to the image of the Hopf invariant n qiu- nn+sk+i(S )-^Ext^u (Λ,Λ), η->-oo. 1 +6 b) The groups Ext '^, (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 Ext^U (Α, Λ)- Extl:h*o{kO'(P),kO'(P)) t Β γ 1; U 1 1i,kO Ά (Zp, Z ) p yk( .( 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. 1 1 2) The homomorphism Ext υ (Λ, Λ) - Ext y (U* (MSU), Λ) was studied. 3) It was established which elements of all these groups Ext1 are realized as images of the Hopf 1 invariant qx. In particular, for the groups Ext '" (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 +2 d3: ΕχΙ^(Λ,Λ)^Εχ^ (Λ, Λ) METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 881 l 2t 3 is nontrivial; it can be shown that d,(E 2' ) = h E\·"'* 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 1 2 that "ΰ2 (Ext ' J) = 0 for i > 4. Analogously for ρ > 2 (see §12). 5. For l £ 8k• + 4, the fact of the following group isomorphism was established: f *" ι ( for I = 4 this fact is false. For t = 8λ + 4, & > 1, it is true and will be proved later (see §11). §11. Cobordism theory in the category S®?Qp Earlier, in §5, it was proved that in the algebra A1 ld r equal to the sum MU Λ: ^j E ^Mv , where ω is not p-adic ji.e., ω = (il7 . . . , i^), all iq 4 p - 1 k for any r, and d (ω) = Σ ί' |. If lp is the Steenrod ring of the spectrum \Ip, where /lp = Φ· -1 · Φ, q ~ 1 then we have: U = s 1) \ ®zQp ^'^ (Λρ) i the appropriately graded ring of infinite matrices of the form (αω . ω .), α ω -ω- ε /Ι ρ , ωι not p-adic and dim (αω , ω .) = 2(/(ω;) — 2ί/(ω;) + dim αΜ.ω. Ii.e., the right-hand side is a constant for the whole matrix and defines the degree of the matrix!. 2) Exty (U*(K), U*(L))®zQp = Extyy (Ul(K), 6'*(L)), where U* = Φ(ί/*®ζ 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 Λρ = Up (Ρ) = Φ U (P) can be chosen to be P polynomials with rational coefficients in the elements xi = [CP "i € Ω^, where the polynomial 1 generator can be identified with LCP''" ] - χΛ = y, in the first nontrivial dimension, equal to ρ — 1. We consider the ring APj; C Λ, generated by the first i polynomial generators y1; .. . , y; ε Λρ . This ring Λρ>ΐ does not depend on the choice of generators. The following fact is clear: the subring Λρ>; C Λρ is invariant with respect to the action of all operations Φ· 52 · Φ on the ring Λρ. 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 Sa! and with respect to Φ, We consider the projection operator Φ; ε Αρ ®ζ) Q such that Φ* : Λρ — Λρί, Φ;|Λρ>Ι = 1 and Φt(y;•) = 0 for / > i. The ring Φ;.4ρΦ; will be denoted by lp>;. It is generated by the operators of multiplication by elements of Λρ>, C A ;; and by operators of the form Φί·Φ·5ω·φ·φί; 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. q The ring Ap is generated by operators of the form Φ· SM· Φ for ω = (&,, . . . , ks), kj = p > - 1. This fact follows easily from properties of the projector Φ and the structure of the spectrum Mp. Qi 9s However, if ω = (p — 1, . . . , p — 1) and at least one q, > i, then clearly σω(Λρ>;) = 0. Hence Ρι qs in the ring Φ^^Φέ it suffices to consider only Φ, · Φ · 5ω · Φ · Φ; for ω = (ρ - 1, ... , p - 1), where all <7-; < i. Additive bases for the rings Ap and Ap>i: a Λ ) Ap = (Λρ ·δω) ,where ω is p-adic and /\ denotes completion (by formal series). ι is t>) ΛΡι1 = (ΑΡι1.8ω)Λ, ω = (ρ' -1,···, p -D,jk We consider the operations ei;^ = S, i. ι (^ times), regarded as elements of the ring APil, i.e., eiik = Φ^α,ΦΦ; € 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(Ap>i.1) = 0; l 3) eji (γι) = ρ, where yi G Ap>j C Λρ is the polynomial generator of dimension p — 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 [yj &t. We have the following Lemma 11.1. a) The subring fej commutes with all operators of multiplication A.piimi C Λρ are f + ei,h • 6i,s = I ) · β j, ;l+ST \ s / ei,h-yqt= 2 el. s-\-m=i Q where e*m{yi)= ( ) 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-· METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 883 with Q>i acting trivially on Λρ^^. 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. 1 In particular, the action of the operators Φ;5ωΦ; for ω = (ρ' — 1, . . . , ρ'* — 1) and for \k < i can also turn out to be nontrivial. k We shall denote ΦΞωΦ by P when ω = (ρ - 1, . . . , ρ - 1) (k times). Ι k k We denote ΦίΡ 'Φί by P . For ρ = 2 we set P = Sqk. As in the ordinary Steenrod algebra mod p, we have here the following fact: the operations Ρ p together with Λρ generate the entire ring Ap (it suffices to take P ). 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 ). fi We now consider the ring Dt, operating on the module Qp LyJ, and the groups Extp (^p LyJ, Qp [yj). We set s>t We consider the groups r <8> Λρ;[Μ. We have s J Lemma 11.2. a) There is a well-defined graded action of the ring Aptl.1 on Sr ' ®ΛΡΐέ., such that: 1)λ(%®μ) = Λ:® λμ, λ, μ€Λρ>ίΜ C Ap>i.,- 2) if eitW =Φ£5ωΦΙ, i ei|(u e Ap, i_i, ω = (p — I,. . . ,p>* — 1), /; < i, then e*u>(x® μ)= 2 β*ω(χ)®β1ωζ(μ), ω=(ω,,ω2) s s here eifO) (Λρ> j.j) is the ordinary action and ej^^x) 6 V (8> Λρ,;-ι for χ €. V , μ 6 APii.l; b) we have the equality Horn A* p .^(^Ip.i-i, Γ··«®Λρ,(_ι) = HomA jiiM (AP,i-i,AP,t^) 5 Proof. Part (b) is obvious. To construct the action of.4p j.jon Γ ® Ap>1., we note that the ring β; acts on Λρ>; = Qp [γ] ® Ap>1.1 naturally, while the action is trivial οηΛρ i_i. From this follows the natural action of the factor-ring APti.l on the groups ExtD. (AP,ir Qp [Vi]) = ExtD. (Qp [y], QP [y]) ® Λρ.ί-ι, where Dt = Qp [yi · &i. It is now easy to derive part (a). 884 S. P. NOVIKOV We note that the ring Β ι is a free right module over &;. We have the following Theorem 11.1. There exists a spectral sequence (Er, dr), where: a) Ea> is associated with Ext/ι (ΛΡ;ί, APji); b) Ev'q coincides with Extp. (Λρ,,-,, Γ" ® A^.J, where Γ" p q + Γ 9 r+ c) dr:E r' -> £P · * ';- a// differentials dr preserve the dimension of elements induced by the dimension of rings and modules; p d) E*-° = Ext 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. 1 For what follows it will be useful to us to compute Ext^' (Qp [γ], Qp lyi). We note that APli= Dl, and the calculation of these groups gives certain information about the ring u ExtA (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 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 2 1 2) ρ (h2k) = /i2k-i, « > 1, x c. Η · , h2k t H < - >(L *& Δ2). METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 885 b) For ρ > 2 the ring Η (C ® Zp) is isomorphic to the ring Zp[x]®A[hu hPi..., hph,...]®Zp(y2, ...,yk,...), where and the BoksteXn hornomorphism β has the following: p i t c) The group Ext ^ (Qp LyJ, Qp LyJ) is nontrivial for t = 2p(p' — 2), 1, ara aP is generated by the following elements: 1) AjX9"1 for p> 2 and all q, 2) h^'1 for q = 2 an 9 1 ? 2 3) Α,χ ' + A2x ' for ρ = 2 and q = 0 mod 2. e) For a/Z f > 1, in the groups H'' * (C; aP : Extc* «?p [y,L 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 ® Zp is commutative, C is obtained from the standard ©(-resolution and G>; ff" (Ci (g) Zp) = Zp [x] (g) Ext^,ΘΖρ (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 [yj · F, 1 Part (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, 1 q 2 and is equal to h^' +. h2x ' for ρ = 2, q = 2s. We shall now prove part (e). Since the homomorphism β is a differential operator, it suffices to l show that H (H (Ci ® Zp), 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 ap (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: } m a) αΓ = β (hhx ), k^ix m ^ 0 .where ρ > 2, where m m 1 b) ah = $(h2kx ) = (h2k-ix + mhzkhiX" ^) 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 · o2m = αϊ + α2 1 2 In particular, we shall denote the element St by h 6 Ext ' . 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})-+ Ext^{V (P), U* (P)) ®z Qp. From Lemma 11.4 and the results of s §7, 8 is derived the following Theorem 11.2. a) For t = 1 the homomorphism γρ is a monomorphism. 1 b) For all ρ > 2 the homomorphism γρ ^ is an isomorphism. 1 2 c) For ρ = 2 the homomorphism γρ is an isomorphism on the groups Ext ' ' 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. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 887 In the formulation of Theorem 11.2 the calculation of the group Ext1'^ +4 ®zQi ls not 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 1 11.1, which converges to the groups Ext^ 2(Λ2>2, Λ22). Namely, we must compute the groups E^' and the differential Λ d2 : Ε2° -+ΕΪ'° « Eiti,;i(A2,,, Λ3ι1) = Er£, (&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 +i y2 = ~ (x2 + (p— i)x£ )L yi = Xi- Ρ We have: •AP.i = Qp[yi], APtz = Qp[yi,y2]. The action of the operation Φ· Ρ · Φ on Λρι and Λρ>2 is given by the formulas: (0, k^p, p + l, Χι k== Φ-Ρ''-Φ{χ2) = Λ ρ ) ' P' As a consequence of this we have Lemma 11.5. The action of the operators Ρ on the generators y2 of the ring Λρ>2 is given by the following formula: where (l2, . . . , l^) is an ordered partition of the integer k and +1 ρ+ ί +ι 1 a) P<(y2)= j(P*(x2) + (p - l)F(xf )) = (Ρ-ΐ){ ι )ρΐχι - for ι φ Pt b) S. P. NOVIKOV l p We note that P (y2) is divisible by ρ for 1 ^ ρ and P (y2) is not divisible by p. ι ε s Now we can describe the action of the ring Ap>l on Γ'® Λρι, where Γ · = Ext^ ' (Qp [y2], P 1 1 Qp [y2]) and Λρ>ι = Qp [yj = Qp [XJ, Xl = [CP ~ ]. The groups Γ '* were computed in Lemma 11.4, 1 ?(p2 l) q flq) part (c). The generator of the group Ext ' " ((?p [y2], Qp iy2}) is obtained as d* (x )/p in the complex Honip (F, Qp [y2]) where F is a D2-ftee acyclic resolution of the module Qp [y2], 9 9 x € Horn* (D2, Qp [y2]) is an element such that % (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 4 by virtue of Lemma 11.5 and ak € <^p Ly,J, α^ = λ yf . From what has been said it is easy to derive 1 Lemma 11.6. The action of the ring APil = Di on Γ ® 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. q Lemma 11.6 follows easily from Lemma 11.4 and the definition of the generators aq = d x /ρ , q q where x C Horn* (F, Qp Ly2J) is such that x (1) = y2 G Qp Ly2J. Further, we compute Horn* (APjl, 1 1 Γ ® Λρ1) = El' in the spectral sequence (E2, d2) of Theorem 11.1, which converges to Ext^* (Λρ>2, Λρ>2); here ΑρΛ = D1 and Λρ>1 = Qp [yj. We have the following 1 Lemma 11.7. The groups Horn*- (ΑΡι1, Γ ® Λρι) are spanned by generators K;J9 of dimension 2 2p' (p — 1) + 2q (p — 1) for all i > 0, q > 0, where the order of the generator Kj>q is p. The proof of the lemma follows easily from Lemma 11.5 and 11.6 by direct calculation. 1 1 Since Horn* (Λρ>1, Γ'(8).Λρ>ι) = /s"' , our problem is to calculate d2 ri"' -^ E2'° = Ext^ (APjl, Λρ>1), where the latter groups are computed in Lemma 11.4 and in the conclusions drawn from it. Direct calculation proves METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 889 1 2 Lemma 11.8. The differential d2 -.El' -> E 2'° of the spectral sequence (Er, dr) converging to Ext4 (^p,2> ^p,2) is 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 l Corollary 11.1. a) For ρ > 2, the kernel Kerd2\E°2 is trivial; 1 b) For ρ = 2, the kernel Ketd2\E2' is generated by elements 4 1 κο,2ί+ι e= HOIUA ^ (ΛΡιι; Γ ® ΛΡΐ1). * > 0. Hence, the image of the homomorphism ;)il ,ι; APil)-)-ExtAp|2 (Λρ,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 s cases ρ = 2 and ρ > 2 is explained by the fact that for ρ > 2 we have h\ = 0 and β (hlx ) = 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 Extl'l (Κ (Ρ), Κ (Ρ)), and this isomorphism is induced by the Riemann-Roch functor λ. ψ b) The Hopf invariant 1 qi: πη+,-ι^—ΕχΙ ·' (*/·(/>), V (Ρ)) A 1 is an epimorphism for t = 8k, t = 8k + 2 and t = 4, and the image lmqi has index 2 in Ext '' for t = 8A: + 6, k > 0, and I = 8k + 4, k > 1. 1 Corollary 11.2. The generators aq of the groups Ext '!' (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). 3 Supplementary remark. It is possible in all dimensions to prove the formula di{a.q) = A · CLQ.2 11 for q = As — 1, 4s + 2, s > 1, where h = al €. Extt ' '^. In particular, for (/ = 4s + 2 this follows from i 4 the fact that h ciq.2 φ. 0 in Ext y, while at the same time o.q_2 is realized by the image of the J- 3 homomorphism, and we must have in £„ that Α ας.2 = 0. *We take this opportunity to note the small computational error in parts (3) and (4) of Theorem 5 of the author's paper L J, which is completely corrected in Theorem 11.3 and Corollary 11.2 of the present paper. 890 S. P. NOVIKOV §12. The Adams spectral sequence and double complexes. Comparison of different cohomology theories We assume that there is given a complex Υ = Υ .ι 6 S and a filtration Y^-Y0^-Yi^...^Yi^..., where the complex of /l*-modules \X* (Yif 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,*+- ···· X such that (a) <^ο?2 = - rf2rf,; (b) ί-> . . . - /V[; Ίΐ /V£.l>;· -> ... I for all / is an ^ -free acyclic resolution and : of the module W;-; (c) if ζ>^ = 2^»3 ^ = ^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 ->...} d is such that Hk (/V,·) = 0 for k > 0, //O(/Vj) is a free /^-module and the complex j . . . Ha(Nk) -i H0(Nk-i -·...! represents an A -free acyclic resolution of the module X (Y). t q As usual, there arises a spectral sequence of the double complex (E r' , 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 1S S (8> zQp meant a set of objects (Zi;), i> — l,/> — 1, and morphisms Z_! _! <— Z_!_ 0 <— Z_ljx <— · • · I 1 i ZQ _X <—Z,, 0 <—ZOii <— • · · ί \ ί t ί ί with the following properties: a) Ζ-,,-i = y, y^ = Z.1j and the filtration Z.lj.1 <- Z.lj0 <- . . . . coincides with the filtration y- y0 -'...- y;- ..." METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 891 b) The filtration Yi I Y%+l — Z~i, i I Z—i., t+i "*— Zo, i I Zo, i+i -*— Zit i I Z\, i+i -<—... represents a geometric realization of the A -free resolution of the module X (Y c/Yi+ t) = Mi, and hence A* (Zft>£/Zj.ji + 1 IJ Zk+lii) = Nkii. afi c) The differentials d1 :Nkti -· !^k-i,i d d2 :Nkti — i^k,i-i coincide with the natural homomorphisms di X* {Zk,i/Zh,i+l [} Zh+l;i)—*~'X* (Zh-l,i/Zk,i [} Z^—l, i+l) , d2 X* {Zh,i/Zk,i+l U Zk+l,i)^>~ X* (Zk,i-l/Zk We make some deductions from the properties of the geometric realization of a double complex: 1. The filtration Z.1(., = Υ <- ZoM «- Zly.1 <-...<- Z; _t <- ... represents the geometric reali- X 2 zation of the /l -free resolution \H* (Vo) ί Η* (,Υ,) «- ... Κ 2. The filtration Y «- Z.t Β |JZO .,<-... U Zj ; *- . . . represents the geometric realization i+j=ft-i of the A -free resolution where d = dl + d2. 3. The double complex (Z) defines two Adams spectral sequences: a) the Adams spectral sequence ΕΓχ in the theory X , induced by the filtration y "*~ ^o,-i U •Z-i.o -«—...-«— U Ζ,, j -«—... with term £* = Ext4 ^ (A"* (Y), A* (K)) for any /( € S; b) the spectral sequence Er of the filtration with term £, = iHom* (K, Yi/Yi + i)\. In view of the presence of the double filtration (Ζί;·) of the complex Υ in all terms of both Adams spectral sequences there arises yet another filtration: in the first case it is equal to φ(χ), x € E^, where φ(χ) coincides in E2 with the filtration in Ext* χ (Χ (Υ), Χ (Κ)) induced by the non-free resolution X (Y) *~ Mo — Mx •- . . . , and in Ε^ is induced by the geometric filtration U Zi,} =5 ... => U . Zu = ...=> Z_ilft. 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. x b)X = H*( ,Zp),A = A. or c) The filtration 7 3 Yo D Y, 3 . . . is an acyclic free filtration in the theory U ® zQp in the theory U C U ® zQp· By virtue of the exactness of the functor U -> Η ( , Z ) in the category D , p p pp) p the filtration Υ D Yo 3 ... is also acyclic (although not free) in the theory X = Η ( , Zp). 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 { , Zp), a Up-free acyclic filtration an< an Υ 3 Yo D Y, I) . . . in the category S ® zQp ^ J X -free acyclic double complex (Z) are homotopy invariants of Υ, and we have the following inequalities: Χ x Ύ Χ iu*p (x) ^ φ(ζ) «ξ ΪΗ*( , ζρ)( ) < iu*p ( ) + Ρ( ), where i = irj*, ίχ = Lt, ζ. are the respective filtrations in the theories U 0 zQp and Η ( , Zp). 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. or ίη c) Both Adams spectral sequences Er in the theories X = II ( , Zp) and U ® zQp ( Up) our case preserve, respectively, the filtrations φ and Φ. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 893 d) The Adams spectral sequence in the theory [I ( , Zp) is such that each differential dr for r > 2 raises the filtration φ at least by 1, i.e.. For the proof of (a) we note that the Up-filtration Υ 3 }'o 3 }\ D . . . depends functorially on the Up- free resolution and is uniquely determined by it. For a fixed Up-filtration the same thing is true with respect to the double complex Ν and the double filtration (Z) defined by it. Parts (b) and (c) are obvious. Part (d) follows immediately from the fact that the complex Υ JYi + l is a direct sum of spectra Wp of the theory Up up to suspension. For such objects the Adams spectral sequence has zero differentials for r > 2, as was proved by Milnor and the author L ' J. The lemma is proved. We now consider the graded ring Λρ C Ωυ ® zQp, where Λρ= Qp [xu . . . , %,,...], dim*! = 2p'- 2. The ring Λρ is a local ring: it has a unique maximal ideal m C Λρ such that Ap/m = Zp. Hence the t i+i bigraded ring Λρ = ^j m /m is an algebra over Zp, and Λρ = Zp [h0, h^ . . ., h^ . . .], where h0 is ρ <—0 associated with multiplication by ρ and dim A; = (1, 2p[ — 1), i.e., hi £ m/m2. Clearly f]ml = 0 and, by [15' 17' 18], 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 onAp is defined by multiplication; 2) the action of Ρ on Λρ is defined so that k l Pi>'(/n)= hi-i and Pi(ha) = 0T / 5=s 1, P (ab) ~ ^ P (a)P*(b). l+s=k i We consider the ring A associated to lp by the filtration Ap D mAp D . . . Γ.· m Ap ?).... We note that in the ordinary Steenrod algebra A there is a normal (exterior) subalgebra Q C .1, Q = A(Q0, . . ., Qi, . . .), dim^t = 2p' - 1, such that A/./Q is isomorphic to the quotient Α/βΛ [J Αβ and Ext/i (//* (Mp, Zp)) = ExtQ(Zp, Zp) = Ap = Zp [k0, hl?..., hi...A. From the results of § 11 and the structure of the Steenrod algebra .1 follows Lemma 12.4. The algebra A associated to the ring Ap is isomorphic to (Ap · A//Q) , where the commutation law ah = %a* (h)al is given by the action of A//Q on Ap defined by the formulas pr h P (hr) = /trM, r > 1, P (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: f t/ (Λρ, Λρ') = Extiz/Q (Zp, Aj) = Exti//Q (Zp, ExtQ (Zp, Zp) t t + l (here, t is the dimension in Ap defined by the filtration Ap = m /'m ). 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 J Extj'(L, V) = EXI'AJ/Q (M, ExtQ (Zp, 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(Zp,Zp)). These sequences have the following properties: 1) In the first, which converges to Ext^ (M, Zp), we have 2) In the second, which is induced by the filtrations in Λρ, Ap , L and which converges to Ext . υ (L, Λ_), we have: Λρ 3) d, = dl and £/•' = £/·' = Εχι*//ρ (i/, Ext^ (Zp, Zp)). q s 9 4) In both spectral sequences there is yet another grading Ep = ^)j Ey ' and Er' '= ^j E r' ' ', 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, Zp); 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, Λρ), 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 ~ s,t',q 2 &co is associated with s+t=m b) The group 2ll -^οό ' Q Q ls associated with ExtTup (L, ΛΡ) = Extfu (U* (Y), U'(P)) ®zQPt where L = U*p(Y),Ap = U*(P). Thus, in the groups tlQ t = Ext2J/Q(M, ExtQ(ZP, Zp)) METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 895 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 4p . Up to this point there has been no difference between ρ = 2 and ρ > 2, if we speak of the results of this section. However, the following theorem shows the comparative simplicity of the case ρ > 2. Theorem 12.1. For any ρ > 2 and complex Υ C D p, the spectral sequence (Er , dr) has all differentials dr - 0 for r > 2. The groups s+i=m s+t=m g are isomorphic to Ext™' (;l/, Zp), where Μ == Η* (Υ, Ζρ), (Zp, Zp) = Zp [h0, ...,hu...l dimfti = (1, 2p* - 1), p and the algebra A//Q generated by the Steenrod powers P acts on ExtQ (Zp, Zp) in the following pi k way: P (hl + i)=hh P (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 s l q ? q spectral sequence" (ΕΓ dr),where E 2- - = Ext^g (,tf, ExtQ(Zp, Zp)), iAe group 2 •#!' ' ' = E?' 9 s q + 7 is associated to Ext™' (1/, Zp), dT:E T -> Ef + r-i, ? + r-i^ an(^ the group 2. ^ca ' is as- s sociated to Ext ·}, (UZtY), Ut(P)), Μ = //*(Y, ZD). •4P We prove Theorem 12.1. In the Steenrod algebra 4 for ρ > 2 there is defined a second grading — the so-called "type in the sense of Cartan," equal to the number of occurrences of the homomorphism β in the iteration. We shall denote by τ (a) > 0 the type of the operation a £ A, with 4=2 ^ττ τ T T T +T2 where τ is the type and A » · <4 2 C 4 i · By the same token, for any Υ Ε Dp there is an extra grading — the type r — in the groups Ext/i (M, Zp), and ϊ (M,ZP)= h where / — r s 0 mod 2p — 2. We note that for Q CC 1, r(()r) = 1 and r(P ) = 0. It is also obvious that r(hi) = 1, Λ, € ExtV, (Zp, Z2), and the type is an invariant of the spectral sequence (dr, Er) for r > 1. 896 S. P. NOVIKOV Since the type is trivial on the ring A//Q, and A//Q C A, all dT - 0 for r > 2, since on the groups s Ext A'^Q (Zp, Ext^ (Zp, Zp)) the type r = ί and r(dry) = r(y) for r > 1. This implies the isomorphism q ExC (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 1 11 l q il<7 12.1 is canonically isomorphic to the sum Έ,Ε^' ' , where El' ' = Ext^' (M, Zp), t is the Cartan 9 9 type, Μ = Η* {Υ, Zp) for Υ € Dp, and 2 Extf' (Λ/, Zp) = Ext™' (M,ZP). In this spectral sequence Ύ . J7S, t, q 'j7s+1> t+r-1, 7+r-l and the group 2 E^''9 is associated to Ext5'' (U* (Y), Up(P)). A t~q=l 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: q 9 1) ET- = 2 1|· '• = Εχί%· (Μ, Ζρ); l q 2) if for some γ €. El' ' 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; 9 3) t'/y'e Εχ$' (Μ, Zp) fs suc/t i/iat ^(y") = 0 for i < k and φ(ά^γ) > φ(γ) + I, then for the pro- y 9 note that for elements J£2J Ext^'' (Μ, Ζρ), φ (y) ^> a). 1 The groups Ext '! (Up (P), Up (P)) were computed in previous sections; they are cyclic for Ά ρ s = 2k (ρ — 1) of order P'* ', where f(k) — 1 is the exponent of the greatest power of ρ which divides A;. 1 (i O Corollary 12.3. The generator ak of the group Ext '!* " ('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, n i > 2, and there is an associated element ak Ε nt (S ); we have φ(αιί) = k — f(k), iji (a^-j _ 1, i^ (a^.) < k - f(k) + 1. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 897 1 Proof. As was shown in ^11, the homomorphism Ext' (Api, Ap>1) — Ext y (Ap, Ap) is an epi- morphism for ρ > 2. For the ring Dr and the module Apil — Qp lxlJ the ring C -- Horn (l· , Apl) was determined (see Lemma 11.4), where φ(χ) -•• 1, φ (hi) = 0, φ (ρ) ~- 1 and d(xh)^=^ Σ f ) \ / / The element a.^ was represented by aj. = (l/p'' )d(xk). From this we have: • · Γ / * \ , · , 1 cp(a ) = ininy φ( . )+J + k —j — f(k) \ = k — f(k) ft I \ j ! J x ι Thus, the filtration φ of the element ak is equal to k — f(k), since the filtration φ is induced by the filtration in the ring Ap. The Corollary is proved. s As is known, the groups Ext^' (Zp, Zp) are equal to Zp for s = 1 or s = 2p' (p — 1) and are gen- 0 1 1 erated by elements it-, / > 0, of type 0 for s - 2p' (p — 1) and h0 £ Ext » ' of type 1 in the sense of Cartan. 0 2 (p 1 1 9 Hence, u, £ Ext^ ' "' -° (Zp, Zp) and h0 £ Ext"' · ^,, Ζρ), where ExC = 2 Exti'" 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 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 Ext/1 (M, Z2). 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 Ext^^ (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 2 .. . , Aks) and is given by the relations Sq uw = 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), 2 2 where ui is a generator of Οι(Μω) and αίίί; = Sq !*;-!. The action of Sq on Λ2 was indicated earlier: £+ 1 2 A2 = Zz[h0,...,hi,...], dimhi = (1, 2 - 1), i > 0, while S? /;, = h0. There follows straightforwardly (by direct calculation) Lemma 12.5. Ext^**0 {Μω, Λ2) for ω - (0) has a system of multiplicative generators: 0 1 1 1 0 2 0 1 2 44 1 0 3 6 AosExt · · , χι e Εχί · · , A4 e Ext · · ' - , i > 2, y^Ext · · αίΐί/ is given by the relation hoxi = 0. s t q We note that the dimension of 'Ex.t ' ' 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,/ 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). METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 899 The proof of Lemma 12.6 for ET follows easily from the calculations L J for Ext . (,!•/, ΖΛ. For the case (Er, dr), part (b) of Lemma 12.6 follows from the fact that the elements x, hl+2 must be zero in Ext υ (U* (U* (MSU), Λ2) on the basis 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 1 Ext v (see Corollary 12.3). Appendix 1 On the formal group of "geometric" cobordism (Theorem of A. S. Miscenko) 7 2 We consider an arbitrary complex X, the group U (X) and its subgroup Map (A , MUx) C U (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 l(t + l) l + i > where xt- € A' '' = Sl\f '' ' 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 = ICP1]. Thus, we have a commutative formal group with graded ring of coefficients Λ, and dim u = dim ν = 2. As is known, the structure of such a group is completely determined by a change of variables l+i g over the ring Λ <£)? Q, "• — g(u) = Z_\l/iU , y0 = 1, such that iSsO g(u®v)=g(u) We have the following 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 ' n 2 xn = L CP ] £ Λ" ", 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 W W bundles over CP χ CP. This multiplication induces a diagonal Δ : U* (CP ) - U* (CP ) 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 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 5ω 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' = 2j Ai w . U = 2j 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 METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 901 It obviously follows that for α. φ 0, β Φ 0 we have: Σ (<* Since μ<° = 0 for all i > 2, μ<° = 1, At = 1 and σ*4) + (t - AjA^ = 0, /t > 1, we have S (σ(ίι)λι + (i — &)λί_^) μ/ = 0 i for all /' > 1, and since ^j μ;· Xs = 6j, we have i 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. n n k Further, since σ* [CP ] = - (re + 1) [CP ' } (see §5, Lemma 5), it follows that \l = xt.,/i, 2; Xj = iCP'i € Λ' 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: y k b) 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 (*) = •Σ *+1 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"(*). U Hence, always Ψ*° ψ£, = ψ*' in A ® 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: m αψ£ = k Wu "a. U 2 ω Proof. Let am = S(m) = Ε A and u € V(CP™) C ί/ (ίΡ ). Then am(u) = 1 f u@...®u since ιιφ... ®aC 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), as in /(-theory. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORD1SM THEORY 903 1. The semigroup of multiplicative operations a £ A ® χΚ, where Δα = a ® a £ A ®Λ A ® z/£. 2. The semigroup of multiplicative operations of dimension 0, U AKCZAKCIA ®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, 2 Just as earlier, a multiplicative operation a £ Αχ is defined by a series a(u), u € U (CP )®Z K the canonical element, α (it) £ Αχ 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, 8 since the series «Ψ is divisible by £ and Ψ[, (u) ς (/* (CP) ® ZK. We have the following 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: α α α α 11; <υυιψ υ2 -~ φ υ ι 2 2ί 2ί 2) Ψ^'* :Λ-Κ - Λ"κ is multiplication by a'. 3) Ψ^ ° a = α-'αΨ^, wAere ο 6 .4^® ZX 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, 0 Ψ (Κ) ~ Up for Κ = Qp, Ψ (Ζ) = Z2, which coincides precisely with the center Ζ κ of the semigroup A ^ 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- U position of series ax · a2(u) takes into account the representation of A ® Κ on Λ ®K. Hence AK is not a group (as usual in formal series of this kind), but a semigroup. An example of a "noninvertible" element α £ .4^ is given by the series where Φ2 = Φ and Φ* (y) = 0, γ £ 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 1 is easy to see that A; = k[ and α = Ψ^ , 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- J 1 tors of Sx (K ) ate in one-one correspondence with the cells of K . We consider the pair (K1, K1'1), where K'/K1'1 is a bouquet of spheres Si\/. . . \/Sl . We adjoin 1 1 1 1 to Sx (K ' ) free generators «,,..., uq of dimension i. A differential in the complex Sx (Κ ' ) + Λ^) + . . . + A(uq ) is introduced as follows: i) dk = xd, λ e Λ; 1 1 1 2) (9u; = Zj β S^ (K'' ), where z; is such that dzy = 0 in S^ (K ' ) and the homology class L z; J 1 1 l 1 1 6 A"* (X * ) is represented by the element equal to <9u;, where d : X* (S[ \/. . . ν S q .) -> A"* (K ' ) 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? METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 905 Now let X = U. Conjecture. For a pair Yl, Y2, the complex Sy (Y1 χ Υ2 ) is homotopically equivalent to the tensor product Su{Yi)®iiuSu{Y2). Let A be an arbitrary fl(/-module. The homology of the complex Sy&o ^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 dim* = 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(xm) = 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. Conjecture. 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 - Tor*^ (U* (Y), A) which converges to ί/*(Υ, A). 1 Theorem 2. Since the complexes Sy (Y)(g) Ω.,Zlx\ = S/C(Y) and Sy (Y) (gjjj Zlx, ar 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, 1 1 2) 0-»-*. ®z[I]Z[a:>x- ]->^.-»-Torz[*r](A;.4Z[a;,ar- ])->0, 3) 0-*-Λ where 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 is possible to find a number of other formulas connecting A*, A , Κ*, Κ , //*, // and also Kunneth formulas for the direct product Υ, χ Y2, starting with the complex Sy ®n Ζ L%J as a Zlxl- module and the fact that Ζ [χ] is one-dimensional, as is Ζ. We note also that Hom^^Z Lac], Ζ [χ]) = Ζ [y], where dimy = - 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] ? n 2. How to compute in U-^(BG) the following elements: let the group Zn act on C linearly, and a without fixed points on C \0, i.e., by means of diagonal matrices (a;/), where aI; = exp(2nixj/m) 2 1 and Xj is a unit in the ring Zm. Then an action of Zm on S "" is defined, and by the same token a 2n i map f%i, . . ., xn : S ' /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 an(l, . . . , 1) ("geometric bordism") and to show that an (xu . . . , χη) φ 0 for all (invertible) xx, . . . , χη Ε Zm (see [ ]), ναη {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, 27Tl m 27Tl m pi = \l \, . . . , pk = \l I, . . . , pm ί are one-dimensional, while as a ring a generator is m pi = ρ with the relation p = 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). METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 907 d) There exists a natural generator u £ k2 (BG) such that every element of It' (BG) has the form ^ y"} U 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). 1 / τη \ We denote the expression πιΨ" (u) by Fm(u) = 2 ( M~ y)* '"*· From Lemma 1 follows Lemma 2. The cell complex S^ (BG) = Hom^^^S^, Zlxi) of modules over Ziyi is a ring with multiplicative generators (over Ζ lyi) ν (of dimension (\))andu (of dimension (2)) and additive basis \ysun, yqvu1}. The differential d in this complex satisfies the Leibnitz formula, commutes with 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 = Hom^r -,(Z lyi, Zlxi), is a complex of free modules. Lemma 2 follows easily from Lemma 1 and Appendix. 2 We turn now to i/*- and U -theories. For the element u €. V (X) = Map(/Y, Μϋγ) C U (X), the series m!Vm,(u) = g-*(mg(u)) (see Appendix 2), where g(u) = 2 iCPn]un+l/(n + 1) is the "Miscenko U 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'21 = 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 = Zm 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 2 1 linear action of the group Zm on 5 "" 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). n 1 Let m = p , ρ a prime and mt = p"' . 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 αη(χυ·,..., 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· METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 909 It is easy to show that for any (xi, . . . , xn) €. Zm χ ... χ Zm 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 (^Zm). We denote υ tne by 2π-ΐ ^- ^2η·ι(^) so-called "geometric bordism" an(l,.. . , 1). In the complex Sy(BG) de- 1 1 scribed in Theorem 1, the element i^2,1.1 is adjoint to vu"" € Sy(BG), i.e., (i>2n.,, i^u™" ) = 1, (;.';„.„ 1111»-' + ') = 0 for A; > 0, where χ 6 Λ*. We shall calculate the function an (x,, . . . , xn) by the following scheme: 1) Clearly, a, (x) = xt,, C f/1(SG)-Zm. tn I 2) If Zj «fti^iji · · · 1x hi) ^ 0, 2j an-k(i/ft4-l,j. · . · , i/n,j) ^ 0, then we have the equation: 3=1 j=i a ZJ n(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 π (7\, Γ2): Λ/ χ Λ/ " , where fixed points (and their orders) correspond to each other. 3) If λχ : BG -> BG is induced by multiplication by χ £ Zm, then an(x, . . . , x) = kx>t (v2n.l), where the structure of λχ „ 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: l "^ [CP ] 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; 1 ft3 (#1, Z2, ^3) = (a-'l^2^3) Vs + ' · [CP ] VS. 910 S. P. NOVIKOV n Suppose given a group of quasicomplex transformations Zp : M -> M" with isolated fixed points Pu . . . , Pq at which the generator Τ 6 Zp has orders %l;, . . . , xnj G Zp, j = 1, . . . , q, where x^j € Ζ x We consider the point (ΧΙΛ, • • • , Xkj, • • • > nq) G Zp" up to a factor μ C Zp, μ φ 0. Thus, {χίΛ, • • • , xng) qn l qn G P ~ m xhe group Sn χ Sq, where S^ is the group of permutations of k elements, acts on P ~ . n Definition. By the type of the action of the group Zp on M with isolated fixed points we shall mean the set of orders of (xli, • • • , Xkj, • • • , Xnq), of any generator Τ G Zp, considered in the pro- qn 1 jective space P ~ j factored by the actions of the group Sn of permutations of orders of each point and the group Sq of permutations of points. From Theorem 3 follows the n qa l Corollary. For p> η and for η = 2,3, the set of types of actions of Ζp on M is given in P ~ by the set of equations: Σ π, IT. • τ Λ — 0 -r- =iL· Π h 9 "ί η is where the xsj are the orders at the point Pj and o^iXij, • • • , xnj) 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), Zp) is bigraded: 1 = Σ/4 ' , where dim = k + β and β is the type. A Conjecture: I) Let the bigradings H{X, Zp) = E// '^and Η (Υ', Ζρ) = I/Z^be 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', Zp) 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. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 911 III) The cohomology A -module // (A, Zp) is Jjigraded, if in // (Α, Ζp ) the bigrading is well defined. IV) All these properties are fulfilled in the subcategory Sgr C S 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), Zp) for X Ε Sgr is arranged as follows: d • Fs'h- β_>_ Fs+r' h' pifr~1 01 We note that h0 Ε Ext^ (Zp, Zp) 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 xt for Υ we have: dq (v ) = h^u , where u C E °/ and ν € Ext/, since 77* (Υ) = Ζ q. From our point of view the situation is thus: 0 1 0 0 1 a) H**(X, Zp) Α/Αβ(α) + Α/Αβ(ν), where u €. //°'°, ν € Ζ/ ' . Hence u* C Ext/' , ν* Ε Ext ' 9 9 0 9 1 9 1 9 1 9 1 and hoU G Ext ' ' ; by dimensional considerations, dq (ν ) Ε Ext ' ' " ,, anandExd Extt ' ' ' = 0. 0 1 0 0 b) #**(Y, Zp) = A/A/3(u) + Α/Αβ(ν), u 6 H°'°, ν € Η · , then υ* € Ext /' and v* € Ext°/'\ 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 Ext1'^ are realized by elements of πίρ) (SN) of the same p) N 1 order; moreover, nl (S ) = Ext^ + ..., where Ext '* = /*® 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 2 2 0 2p 1 2p2(p l) + 2p 2 χ € Ext '^ ^" where d^.^x) = ?, since Ext J' ' ' " ^ Q_ In realitV; these elemerits in (/-theory are "inherited" from ordinary cohomology theory Η ( , Zp) 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 tr. 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** (Zp, Zp) and converges to Ext**^ (Λ, Λ) 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 BIBLIOGRAPHY LlJ J. F. Adams, On the structure and applications of the Steenrod algebra, Comment. Math. Helv. 32 (1958), 180-214. MR 20 #2711. [2] , Vector fields on spheres, Ann. of Math. (2)75 (1962), 603-632. MR 25 #2614. [3] , On the groups J (X). I, Topology 2 (1963), 181-195. MR 28 #2553. L4J J. F. Adams and M. F. Atiyah, K-theory and the Hopf invariant, Quart. J. Math. Oxford Ser. (2) 17 (1966), 31-38. MR 33 #6618. [5] M. F. Atiyah, Bordism andcobordism, Proc. Cambridge Philos. Soc. 57 (1961), 200-208. MR 23 #A4150. [6] , Thorn complexes, Proc. London Math. Soc. (3) 11 (1961), 291-310. MR 24 #A1727. L7J , Characters and cohomology of finite groups, Inst. Hautes Etudes Sci. Publ. Math. No. 9 (1961), 23-64. MR 26 #6228. L8J M. F. Atiyah and F. Hirzebruch, Riemann-Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc. 65 (1959), 276-281. MR 22 #989. L9J D. W. Anderson, E. Brown and F. Peterson, SU-cobordism, KO-characteristic numbers and the Kervaire invariant, Ann. of Math. (2) 83 (1966), 54-67. MR 32 #6470. [l0] , Spin cobordism, Bull. Amer. Math. Soc. 72(1966), 256-260. MR 32 #8349. [llJ P. Conner and E. Floyd, Cobordism theories, Princeton Univ. Press, Princeton, N. J., 1963, pp. 1-15. Preprint. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 913 [lH , The SU-bordism theory, Bull. Amer. Math. Soc. 70 (1964), 670-675- MR 29 #5253- [13] , Torsion in SU-bordism, Mem. Amer. Math. Soc. No. 60 (1966), 1-73. MR 32 #6471. [14] M. Kervaire and J. Milnor, Bernoulli numbers, homotopy groups and a theorem of Rohlin, Proc. Internal. Congr. Math., 1958, Cambridge Univ. Press, New York, (I960), pp. 454-458. MR 22 #5537. LI5J J. Milnor, On the cobordism ring Ω and a complex analogue. I, Amer. J. Math. 82 (I960), 505-521. MR 22 #9975. [l6J A. S. Miscenko, Spectral sequences of Adams type, Mat. Zametki 1 (1967), 339-346. (Russian) MR 35 #2286. 117] S. P. Novikov, Some problems in the topology of manifolds connected with the theory of Thorn spaces, Dokl. Akad. Nauk SSSR 132 (I960), 1031-1034 = Soviet Math. Dokl. 1 (I960), 717-720. MR 22 #12545. [lal , Homotopy properties of Thorn complexes, Mat. Sb. 57 (99) (1962), 407-442. (Russian) MR 28 #615. [19J , Rings of operations and spectral sequences of Adams type in extraordinary cohom- ology theories. U-cobordism and K-theory, Dokl. Akad. Nauk SSSR 172 (1967), 33-36 = Soviet Math. Dokl. 8 (1967), 27-31. MR 36 #884. [20] V. A. Rohlin, Intrinsic homology, Dokl. Akad. Nauk SSSR 89 (1953), 789-792. (Russian). MR 15, 53. [2l] , Intrinsic homology theory, Uspehi Mat. Nauk 14 (1959), no. 4 (88), 3-20; English transl., Amer. Math. Soc. Transl. (2) 30 (1963), 255-271. MR 22 #11402; MR 27 #5252. L22J R. Stong, Relations among characteristic numbers. I, Topology 4 (1965), 267-281. MR 33 #740- [23] C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. 72 (I960), 292-311. MR 22 #11403- i.24] S. P. Novikov, Cohomology of the Steenrod algebra, Dokl. Akad. Nauk SSSR, 128 (1959), 893-895 (Russian) MR 22 #1889. L25J J. Cohen, Some results on the stable homotopy groups of spheres, Bull. Amer. Math. Soc. 72 (1966), 732-735. MR 35 #1022. th I26J E. Brown and F. Peterson, A spectrum whose 7.p cohomology is the algebra of reduced p powers, Topology 5 (1966), 149-154. MR 33 #719. 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 the cohomology of a Hopf algebra with commutative diagonal. It turns out that for all ρ > 2 these "Steenrod powers" Stp1 are defined and nontrivial for i = 0, 1 mod (p-1), i > 0. These peculiar operations have never been noted in more recent literature on these questions, although they are of value; for example, they reflect on the multiplica- tive formulas of Theorem 2 in L^J for ρ > 2. METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 839 zQp is °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." Q) by means of the operator of multiplication by ys, and their defining spectra differ only by suspension. 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- 1 bers σ* (y) € k = p — 1. Moreover, a complete system of orthog- (0 (ι) (ί) (ί) onal projectors Φ was constructed, ΣΦ = 1, φ ·Φ<'> = 0, ι'Φ j, where the Φ (U* Z2) i/ie kernel Ker /3 coincides with the image Im Hence, the homomorphism of reduction modulo p, Zp has a system of generators [I ;i, whileZp); 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 are "•» ($") ®zQp associated to Ext L, (U (P), V (P))&zQp- iEquivalently: all differentials dr, r > 2, are zero in the Adams spectral sequence over Vp.\V=U-\-V-\- 2 xi,juivKzQp obtained from K(Zp) inductively by means of bigraded morphisms and passage to "kernels" and "cokernels"