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The Methods of Algebraic Topology from the Viewpoint of Cobordism Theory

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The Methods of Algebraic Topology from the Viewpoint of Cobordism Theory Home Search Collections Journals About Contact us My IOPscience THE METHODS OF ALGEBRAIC TOPOLOGY FROM THE VIEWPOINT OF COBORDISM THEORY This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1967 Math. USSR Izv. 1 827 (http://iopscience.iop.org/0025-5726/1/4/A06) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 18.7.29.240 The article was downloaded on 06/08/2012 at 22:38 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 <g) z Qp D A V, and Λ ® ζ Qp is a local ring u with maximal ideal m C Λ ® z Qp, where h ® z Qp/m = Ζp. Note that Λρ is an A ^-module and A is also a left Ap-module. Consider the following rings: mv = m Π Λρ, Λρ / mp = Zp, *It may be shown that the Adams spectral sequence is the generalization specifically for S-categories (see § 1) of "the universal coefficient formula," and this is used in the proofs of Theorems 1 and 2 of Appendix 3- **f/*-theory is a direct summand of the cohomology theory U* ® Qp, having spectrum Mp such 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) <g> zQp- We now recall the striking difference between the Steenrod algebra modulo 2 and modulo ρ > 2. As is shown in H. Cartan's well-known work, the Steenrod algebra for ρ > 2 in addition to the usual grading possesses a second grading ("the number of occurrences of the Bokstetn homomorphism") of a type which cannot be defined for ρ = 2 (it is only correct modulo 2 for ρ = 2). Therefore for ρ > 2 the cohomology Ext/i (Zp, Zp) has a triple grading in distinction to ρ = 2. In §12 we show: Lemma. There is a canonical algebra isomorphism Ζ,*** -η ,Τ Τ ν τ-ι *** = Ext^u (ΛΒ, Λρ) = ExtA (ΖριΖΒ) for />>2. From this it follows that the algebra E2 for the "algebraic Adams spectral sequence" Er is not associated, but is canonically isomorphic to the algebra Ext^iZp, 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).
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