Replacing homotopy actions by topological actions. II

Abstract

A homotopy action of a group G G on a space X X is a homomorphism from G G to the group HAUT ( X ) {\operatorname {HAUT}}(X) of homotopy classes of homotopy equivalences of X X . George Cooke developed an obstruction theory to determine if a homotopy action is equivalent up to homotopy to a topological action. The question studied in this paper is: Given a diagram of spaces with homotopy actions of G G and maps between them that are equivariant up to homotopy, when can the diagram be replaced by a homotopy equivalent diagram of G G -spaces and G G -equivariant maps? We find that the obstruction theory of Cooke has a natural extension to this context.