J. F. Adams

On the structure and applications of the steenrod algebra

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On Chern Characters and the Structure of the unitary group

The purpose of this paper is twofold. In order to state our first aim, let U denote the ‘infinite’ unitary group, and let BU be a classifying space for U . Then Bott (2),(3) has shown that we propose to investigate the ‘Postnikov system’ of BU .

Vector fields on spheres

The question of vector fields on spheres arises in homotopy theory and in the theory of fibre bundles, and it presents a classical problem, which may be explained as follows. For each n, let Sn-I be the unit sphere in euclidean n-space Rn. A vector field on Sn-1 is a continuous function v assigning to each point x of Sn-1 a vector v(x) tangent to Sn-1 at x. Given r such fields v1, v2, ..., Vr, we say that they are linearly independent if the vectors v1(x), v2(x), *--, vr(x) are linearly independent for all x. The problem, then, is the following: for each n, what is the maximum number r of linearly independent vector fields on Sn-i? For previous work and background material on this problem, we refer the reader to [1, 10, 11, 12, 13, 14, 15, 16]. In particular, we recall that if we are given r linearly independent vector fields vi(x), then by orthogonalisation it is easy to construct r fields wi(x) such that w1(x), w2(x), *I * , wr(x) are orthonormal for each x. These r fields constitute a cross-section of the appropriate Stiefel fibering. The strongest known positive result about the problem derives from the Hurwitz-Radon-Eckmann theorem in linear algebra [8]. It may be stated as follows (cf. James [13]). Let us write n = (2a + 1)2b and b = c + 4d, where a, b, c and d are integers and 0 < c < 3; let us define p(n) = 2c + 8d. Then there exist p(n) 1 linearly independent vector fields on Sn-1. It is the object of the present paper to prove that the positive result stated above is best possible.

A generalization of the segal conjecture

THEOREM 1.4 below is a generalization of the Segal conjecture about equivariant cohomotopy. It asserts an invariance property of the G-cohomology-theory S- ‘xE(-); obtained from equivariant cohomotopy rrE by first localizing with respect to a general multiplicativelyclosed subset S in the Bumside ring A(G), and then completing with respect to a general ideal I c A(G). We first explain how we place previous “localization theorems” and “completion theorems” in one setting by formulating suitable invariance statements. Let G be a finite group; all our G-spaces will be G-C W complexes [23]. Let # be some class of subgroups and letf: X+ Y be a G-map. We will say thatfis an “&“-equivalence” if the induced map of fixed-point-setsfH: XH + Y H is an ordinary homotopy equivalence for each HEX. (Thus we may assume without loss of generality that Z is closed under passing to conjugate subgroups.) Let h be a functor defined on G-spaces and G-maps; we will say that h is “&?-invariant” if it carries each x-equivalence to an isomorphism in the target category of h. The same property was previously introduced in [34] and studied further in [35]. In particular, let 2 be the class of all subgroups H c G; then an &?-equivalence is just a Ghomotopy-equivalence, and every G-cohomology-theory is Z-invariant. To place “localization theorems” in this setting, we assume that 2 is closed under passing to conjugate subgroups and larger subgroups. Then for any X we have an fl-fixed-point subcomplex Xx= u{X~:HEZJ, and the inclusion i:X”-+X is an &?-equivalence.

A GENERALIZATION OF THE ATIYAH-SEGAL COMPLETION THEOREM

KnG (X) = {KnG(X.)}, where X, runs over the finite subcomplexes of X. For a subgroup H of G, we have a restriction homomorphism I~. G* R(G)-+R(H) and we let 1; be its kernel. (Subgroups are understood to be closed.) A set # of subgroups of G closed under subconjugacy is called a family. We let (Kg);, the y-adic completion of Kg, denote the progroup valued G-cohomology theory specified by K:(X)? = {K;(X,)/JK”G(X,)}, where J runs over the finite products of ideals Z

Finite H-spaces and Lie groups

with HEY. (The relevant information about progroups is summarized in [2, 923.)

Buhstaber's work on two-valued formal groups

The definition of an H-space goes back to Serre, in his thesis [7]. The given structure of an H-space comprises three things: a topological space X, a base-point eE X, and a “product map” ,U : Xx X-A’. These, of course, may be required to satisfy various axioms. For present purposes vve don’t need to know the axioms; we do need to know some examples. In the first class of examples, the space X is a topological group G; the point e is the unit element in G; and the map p is given by the product in the group, b(g,h) =gh. The topological groups of most important to us here are the Lie groups. In the second class of examples, X is a loop-space QY. That is, one starts from a space Y with base-point yo; and one forms the space RY of continuous functions LL): [0, I], 0, l- Y,yo,yo. These functions are called loops, and one gives the set of loops the compact-open topology. The base-point e is the loop constant at ,VO; and one defines the product ,U (uJ,, 0’) of two loops in the usual way, that is,

On the Non-Existence of Elements of Hopf Invariant One

Abstract The paper which follows may be regarded as the best substitute available for the lecture which V.M. Buhstaber would have delivered to the International Congress of Mathematicians, Vancouver 1974, if he had been present. (We would like to say how sorry we are that he was not able to be there.) In fact, we originally agreed to prepare it for submission to the Proceedings of the Congress. The text is in the form of a report on Buhstabers work by J.F. Adams and A. Liulevicius, and these two authors accept entire responsibility for it. Of course, our primary source is the account of Buhstabers work which we heard at the Congress from A.T. Fomenko, and we would like to thank him for all his help. But we have also tried to improve our understanding by consulting the papers which Buhstaber has published in Russian. We assume that the reader is aware of the connection between complex cobordism and the theory of formal groups [2, 5]; this work is generally respected. The topic of two-valued formal groups represents an extension of this theory. It is conceived partly as a contribution to pure algebra, but it is inspired by an application to algebraic topology; this application lies in the theory of characteristic classes of symplectic bundles, and in the study of symplectic cobordism.