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

Let G be a finite group, let X and Y be finite G-complexes, and suppose that for each KG, Y^K is dim(X^K)-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 F(X, Y). As a corollary, we are able to analyze at a prime p, the homotopy fixed point set of the circle action on X, where X denotes the free loop space of X, and X is a simply connected finite complex.Supported in part by NSF DMS 86-02430.To A. Grothendieck on the occasion of his sixtieth birthday