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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 THE METHODS OF ALGEBRAIC TOPOLOGY FROM THE VIEWPOINT OF COBORDISM THEORY S. P. NOVIKOV Abstract. The goal of this work is the construction of the analogue to the Adams spectral sequence in cobordism theory, calculation of the ring of co- homology operations in this theory, and also a number of applications: to the problem of computing homotopy groups and the classical Adams spectral sequence, fixed points of transformations of period p, and others. Introduction In algebraic topology during the last few years the role of the so-called extraor- dinary homology and cohomology theories has started to become apparent; these theories satisfy all the Eilenberg–Steenrod axioms, except the axiom on the homol- ogy 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 Poincar´etype, Thom isomor- phisms, 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, 5, 8, 9, 11, 12]). These ideas and notions gave rise to a series of brilliant results ([2]–[13]). In time there became manifest two important types of such theories: (1) theories of “K type” and (2) theories of “cobordism type” and their dual homology (“bordism”) theories. The present work is connected mainly with the theory of unitary cobordism. It is a detailed account and further development of the author’s work [19]. The structure of the homology of a point in the unitary cobordism theory was first discovered by Milnor [15] and the author [17]; the most complete and systematic account together with the structure of the ring can be found in [18]. Moreover, in recent work of Stong [22] and Hattori important relations of unitary cobordism to K-theory were found. We freely use the results and methods of all these works later, and we refer the reader to the works [15, 17, 18, 22] for preliminary information. Our basic aim is the development of new methods which allow us to compute stable homotopy invariants in a regular fashion with the help of extraordinary ho- mology theories, by analogy with the method of Cartan–Serre–Adams in the usual classical Zp-cohomology theory. We have succeeded in the complete computation of the analogue of the Steenrod algebra and the construction of a “spectral sequence Date: Received 10 APR 67. Editor’s note: Some small additions, contained in braces {}, have been made in translation. 1 2 S. P. NOVIKOV of Adams type”1 in some cohomology theories, of which the most important is the theory of U-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 extra- ordinary cohomology theory) able to resolve difficulties connected with the stable homotopy groups of spheres? In the author’s opinion, it succeeds in showing some principal (and new) sides of this problem, which allow us to put forth arguments about the nearness of the problems to solution and the formulation of final answers. First of all, the question should be separated into two parts: (1) the correct selec- tion of the theory of cobordism type as “leading” in this program, and why it is richer than cohomology and K-theory; (2) how to look at the problem of homotopy groups of spheres from the point of view of cobordism theory. The answer to the first part of the question is not complicated. As is shown in Appendix 3, if we have any other “good” cohomology theory, then it has the form of cobordism with coefficients in an Ω-module. Besides, working as in §§ 9 and 12, it is possible to convince oneself that these give the best filtrations for homotopy groups (at any rate, for complexes without torsion; for p = 2 it may be that the appropriate substitute for MU is MSU). In this way, the other theories lead to the scheme of cobordism theory, and there their properties may be exploited in our program by means of homological algebra, as shown in many parts of the present work. We now attempt to answer the second fundamental part of the question. Here U U we must initially formulate some notions and assertions. Let Ap [A ] be the ∗ ∗ ∗ ring of cohomology operations in Up -theory [U , respectively], Λp = Up (P ), Λ = ∗ 2 U U (P ), P = point, Qp = p-adic integers. Note that Λ ⊂ A . The ring over Qp, U U Λ ⊗Z Qp ⊃ Λp, lies in A ⊗Z Qp ⊃ Ap , and Λ ⊗Z Qp is a local ring with maximal U ideal m ⊂ Λ ⊗Z Qp, where Λ ⊗Z Qp/m = Zp. Note that Λp is an Ap -module and U Ap is also a left Λp-module. 1It 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 Theo- rems 1 and 2 of Appendix 3. 2 ∗ ∗ Up -theory is a direct summand of the cohomology theory U ⊗ Qp, having spectrum Mp such ∗ that H (Mp,Zp) = A/(βA + Aβ) {where A is the Steenrod algebra over Zp and β is the Bokˇste˘ın operator} (see §§ 1, 5, 11, 12). METHODS OF ALGEBRAIC TOPOLOGY FROM COBORDISM THEORY 3 Consider the following rings: mp = m ∩ Λp, Λp/mp = Zp, ¯ X i i+1 Λp = mp/mp , i≥0 ¯ ¯U X i U i+1 U A = Ap = mpAp /mp Ap , i≥0 where Λ¯ p is an A¯-module. ˜ ˜ In this situation arises as usual a spectral sequence (E˜r, dr), where ˜ ∗∗ ˜ ∗∗∗ ¯ ¯ Er & Ext U (Λ, Λ) ⊗Z Qp, E2 = Ext ¯U (Λp, Λp), A Ap determined by the maximal ideal mp ⊂ Λp and the induced filtrations. It turns out that for all p > 2 the following holds: ∗∗∗ ¯ ¯ ∗∗ Theorem. The ring Ext ¯U (Λp, Λp) is isomorphic to Ext (Zp,Zp), and the alge- Ap A ˜ ˜ braic spectral sequence (E˜r, dr) is associated with the “geometric” spectral sequence ∗ of Adams in the theory H ( ,Zp). Here p > 2 and A is the usual Steenrod algebra for Zp-cohomology. ˜∗∗∗ ∗∗∗ We note that E∞ is associated with ExtAU (Λ, Λ) ⊗Z Qp (more precisely stated ˜ ˜ in § 12). A priori the spectral sequence (E˜r, dr) is cruder than the Adams spectral ∗ ˜∗∗∗ sequence in H ( ,Zp)-theory and E∞ 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 E˜∞ is associated with ExtAU (Λ, Λ) ⊗Z Qp. We now recall the striking difference between the Steenrod algebra modulo 2 and modulo p > 2. As is shown in H. Cartan’s well-known work, the Steenrod algebra for p > 2 in addition to the usual grading possesses a second grading (“the number of occurrences of the Bokˇste˘ın homomorphism”) of a type which cannot be defined for p = 2 (it is only correct modulo 2 for p = 2). Therefore for p > 2 the cohomology ExtA(Zp,Zp) has a triple grading in distinction to p = 2. In § 12 we show: Lemma. There is a canonical algebra isomorphism ˜∗∗∗ ∗∗∗ ¯ ¯ ∗∗∗ E = Ext ¯U (Λp, Λp) = Ext (Zp,Zp) for p > 2. 2 Ap A ˜ From this it follows that the algebra E˜2 for the “algebraic Adams spectral ˜ sequence” E˜r is not associated, but is canonically isomorphic to the algebra ExtA(Zp,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 algebraic computation of the Steenrod algebra A, but has a deeper geometric significance, then it is not out of the question that the whole Adams spectral sequence is not bigraded, but trigraded, as is the term ∗∗∗ E2 = ExtA (Zp,Zp), p 6= 2. 4 S. P. NOVIKOV From this, obviously, would follow the corollary: for p > 2 the algebraic Adams ˜ ˜ spectral sequence (E˜r, dr) coincides with the topological Adams spectral sequence (Er, dr), if the sequence (Er, dr) is trigraded by means of the Cartan grading, as is ˜ ˜ N P s,t (Er, dr). Therefore the orders |πN+i(S )| would coincide with ExtAU (Λ, Λ) t−s=i up to a factor of the form 2h. Moreover, this corollary would hold for all complexes without torsion (see § 12). The case p = 2 is more complicated, although even there, there are clear algebraic rules for computing 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.
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