Stefan Jackowski

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

Vector bundles over classifying spaces of compact Lie groups

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

Homotopy theory of classifying spaces of compact Lie groups

The completion theorem of Atiyah and Segal [AS] says that the complex K-theory group K(BG) of the classifying space of any compact Lie group G is isomorphic to R(G) : the representation ring completed with respect to its augmentation ideal. However, the group K(BG) = [BG,Z × BU ] does not directly contain information about vector bundles over the infinite dimensional complex BG itself. The purpose of this paper is to compare the Grothendieck group of vector bundles over BG, which we denote K(BG), with both K(BG) and R(G). The main result is an algebraic description of K(BG) in terms of the representation rings of certain subgroups of G. As one consequence, we show that of the natural maps

A fixed-point theorem for p-group actions

The basic problem of homotopy theory is to classify spaces and maps between spaces, up to homotopy, by means of invariants like cohomology. In the last decade some striking progress has been made with this problem when the spaces involved are classifying spaces of compact Lie groups. For example, it has been shown, for G connected and simple, that if two self maps of BG agree in rational cohomology then they are homotopic. It has also been shown that if a space X has the same mod p cohomology, cup product, and Steenrod operations as a classifying space BG then (at least if p is odd and G is a classical group) X is actually homotopy equivalent to BG after mod p completion. Similar methods have also been used to obtain new results on Steenrod’s problem of constructing spaces with a given polynomial cohomology ring. The aim of this paper is to describe these results and the methods used to prove them.

A note on the subgroup theorem in cohomological complexity theory

We prove Sullivans fixed-point conjecture for fixed-point-free actions of compact Lie groups which are extensions of a p-group by a torus. Moreover, we show that for a finite p-group G and a compact or finitely dimensional paracompact G-space X the fixed point set XG is nonempty iff the induced homomorphism of zero-dimensional stable cohomotopy groups 7r?(BG) 7r0(EG XG X) is injective. Let qG: EG BG be a universal principal G-bundle. For a G-space X consider the associated bundle qG: EG XG X -BG with fiber X. Let 7r0(-) denote the zero-dimensional stable cohomotopy functor. THEOREM A. Let G be a finite p-group. Suppose that a G-space X is compact, or is paracompact with finite cohomological dimension. Then the following conditions are equivalent: (a) The fixed point set XG is nonempty, (b) there exists a G-map EG C X, (c) qG has a section, (d) the induced homomorphism (qqG)*: 7r?(BG) w7r(EG xG X) is injective. (For a definition of the cohomological dimension cf. Quillen [1971].) Our theorem generalizes results of W. Y. Hsiang and T. tom Dieck. Hsiang [1975, IV.I] proved it on the assumption that G is an elementary abelian p-group or a torus, and with singular cohomology replacing zero-dimensional stable cohomotopy in (d). T. tom Dieck [1972a] extended Hsiangs result to abelian compact Lie groups G such that the group of components G/Co is a p-group, considering in (d) unitary cobordism theory instead of singular cohomology. In both theorems, in the case of a nondiscrete group G and a noncompact space X, we have to assume additionally that X has only finitely many orbit types. Proofs of those results are based on computations of appropriate cohomology of the classifying space of the acting group. Similary the proof of Theorem A relies on G. Carlssons work (Carlsson [1984]) describing stable cohomotopy of classifying spaces. Theorem A is related to a conjecture of D. Sullivan [1970, p. 5.118]. Suppose that G and X are as in Theorem A. The (generalized) Sullivan conjecture says that the map XG mapG(EG, X) which assigns to every fixed point x c XG the constant map f.: EG -X induces an isomorphism of mod p cohomology. For Received by the editors February 6, 1986 and, in revised form, July 2, 1986. 1980 Mathematics SubJect Classification (1985 Remsiotn). Primary 55N25, 57S99.

Homotopy classification of self-maps of BG via G-actions

Abstract A simple proof of the ‘Subgroup Lemma’ for varities of modules is presented thus filling in a gap in an argument of the second author.