On the homotopy fixed point problem for free loop spaces and other function complexes

Abstract

Let G be a finite group, let X and Y be finite G-complexes, and suppose that for each K $$ \subseteq$$ G, YK is dim(XK)-connected and simple. G acts on the function complex F(X, Y) by conjugation of maps. We give a complete analysis of the homotopy fixed point set of the space O8?8F(X, Y). As a corollary, we are able to analyze at a prime p, the homotopy fixed point set of the circle action on O8?8?X, where ?X denotes the free loop space of X, and X is a simply connected finite complex.