On equivariant fibrations of $G$-CW-complexes

TL;DR

This paper proves that equivariant Serre fibrations of G-CW-complexes are also equivariant Hurewicz fibrations within compactly generated G-spaces, extending classical results to the equivariant setting.

Contribution

It establishes the equivalence of equivariant Serre and Hurewicz fibrations for G-CW-complexes and provides a characterization via fixed point maps, solving a problem posed by James and Segal.

Findings

Equivariant Serre fibrations are Hurewicz fibrations in the equivariant setting.

G-CW-complexes can be embedded as equivariant retracts in simplicial G-complexes.

Fixed point maps characterize Hurewicz G-fibrations for G-CW-complexes.

Abstract

It is proved that if $G$ is a compact Lie group, then an equivariant Serre fibration of $G$-CW-complexes is an equivariant Hurewicz fibration in the class of compactly generated $G$-spaces. In the nonequivariant setting, this result is due to Steinberger, West and Cauty. The main theorem is proved using the following key result: a $G$-CW-complex can be embedded as an equivariant retract in a simplicial $G$-complex. It is also proved that an equivariant map $p: E \to B$ of $G$-CW-complexes is a Hurewicz $G$-fibration if and only if the $H$-fixed point map $p^H : E^H \to B^H$ is a Hurewicz fibration for any closed subgroup $H$ of $G$. This gives a solution to the problem of James and Segal in the case of $G$-CW-complexes.