Realizing homotopy group actions

TL;DR

This paper introduces a new framework for modeling and constructing $G$-spaces with prescribed homotopy actions of finite groups, using Bredon homotopy actions and lifting techniques.

Contribution

It defines Bredon homotopy actions for finite groups and provides a method to realize these actions as actual $G$-spaces through a sequence of lifting problems.

Findings

A procedure to construct $G$-spaces from Bredon homotopy data.

A characterization of homotopy actions via fixed point sets and homotopy groups.

Potential applications in transferring group actions along maps.

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 $(X_H)_{H\leq G}$ 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 $X'$ 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 $f:X\to Y$.