On an extension of localization theorem and generalized Conner conjecture

Abstract

Let G be a compact Lie group. Then Borel-Segal-Quillen-Hsiang localization theorems are known for any G-space X where X is any compact Hausdorff space or a paracompact Hausdorff space of finite cohomology dimen- sion. The Conner conjecture proved by Oliver and its various generalizations by Skjelbred are also known for only these two classes of spaces. In this paper we extend all of these results for the equivariant category of all finitistic G-spaces. For the case when G = Zp or G = T (torus) some of these results were already proved by Bredon. 1. Introduction. The Borel-Segal-Quillen-Hsiang (10), (12) localization theorem states: Let X be a paracompact (always assumed Hausdorff) space and G a compact Lie group acting continuously on X with finitely many orbit types. Let X be of finite cohomological dimension over a coefficient ring A. Let S G H*(BG, A) be a multi- plicative system. Then the localized restriction homomorphism S ~ xHq(X, A) —* S~xHq(Xs, A) is an isomorphism. The Conner conjecture as proved by Ohver (11) states: Let X be a paracompact space and G a compact Lie group acting continuously on X with finitely many orbit types. Let A = Z,Zpor Q and assume that X is of finite cohomology dimension over A. Then if X is A-acyclic then so is X/ G. Both of the above results hold when X is compact Hausdorff even without the condition on number of orbit types and on the cohomology dimension of X. However, for paracompact spaces both of these conditions have been consistently and conveni- ently taken, not only in the above theorems and the results proved before them (7), but also in obtaining the generalized Conner conjecture as formulated and proved by Skjelbred (14). In all these cases one can easily see that the condition on number of orbit types is needed. But the condition on the cohomology dimension of X is not really needed except of course in the methods of proofs. Also it should be noted that the condition on the cohomology dimension of X is not, in contrast to the condition on number of orbit types, a condition on the transformation group (G, X). It depends on the coefficient ring. Naturally, therefore, one would like to substitute some condition on the transformation group, preferably a weaker one, in place of cohomology dimension of X to achieve the same results. The objective of this paper is to prove all of the above-mentioned theorems by substituting the 'Swan' condition (4, p. 133), as has been already done by Bredon (4, Chapters HI, VII) in proving several Smith-type theorems for Z^-actions and rr-actions. A