Realizing homotopy group actions

Abstract

For any finite group $G$, we define the notion of a Bredon homotopy action of $G$, modelled on the diagram of fixed point sets $(X_H{)}_{H\le G}$ for a $G$-space $X$, together with a pointed homotopy action of the group $N_{G}H/H$ on $X^H/({\bigcup }_{H<K}{X}^K)$. We then describe a procedure for constructing a suitable diagram $\underline{X}:O_G^{op}\to Top$ from this data, by solving a sequence of elementary lifting problems. If successful, we obtain a $G$-space $X^{\prime }$ 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 $f:X\to Y$.

23 pages