Fixed-points of finite group actions on contractible complexes

Abstract

This is a summary of results obtained while attempting to classify the finite complexes which can be fixed-point sets of cellular actions of a given group on finite contractible CW complexes. Here, by a cellular action is meant one where the action of any group element takes the interior of any cell to the interior of some other cell, and takes a cell to itself only via the identity map. For groups of prime power order, the question has already been answered by P. A. Smith [4] and Lowell Jones [2] : if \G = p, a finite complex can be a fixed-point set if and only if it is Zp-acyclic. The main tools for answering the question for groups not of prime power order are certain functions defined below, which serve as bookkeeping devices for controlling the fixed-point structure of a space with group action. Let G denote the class of finite groups G with a normal subgroup P 2) such that G acts on X with fixed-point set F, and such that Hn(X) is a projective Z[G]-module. To any G-resolution X there corresponds a unique resolving function y satisfying XiX**) = 1 + Z <p(K) for all 0 *H C G, KDH