Realizing homotopy group actions

Abstract

For any finite group $G$, we define the notion of a \emph{Bredon homotopy action} of $G$, modelled on the diagram of fixed point sets \ensuremath{(\mathbf{X}\sp{H})\sb{H\leq G}} for a $G$-space $\mathbf{X}$, together with a pointed homotopy action of the group \ensuremath{N\sb{G}H/H} on \ensuremath{\mathbf{X}\sp{H}/(\bigcup\sb{H<K} \mathbf{X}\sp{K}).} We then describe a procedure for constructing a suitable diagram \ensuremath{\underline{\mbox{X}}:{\EuScript O}_{G}\op\to{\EuScript Top}} from this data, by solving a sequence of elementary lifting problems. If successful, we obtain a $G$-space \ensuremath{\mathbf{X}'}\ realizing the given homotopy information, determined up to Bredon $G$-homotopy type. Such lifting methods may also be used to understand other homotopy questions about group actions, such as transferring a $G$-action along a map \ensuremath{f:\mathbf{X}\to \mathbf{Y}. }