Notes on equivariant bundles

TL;DR

This paper compares two notions of equivariant fiber and principal bundles, clarifies foundational results in equivariant bundle theory, and provides proofs of equivariant generalizations of classical results involving classifying spaces.

Contribution

It clarifies the relationship between different definitions of equivariant bundles and extends classical results to the equivariant setting with detailed proofs.

Findings

Comparison of two notions of G-fiber bundles and G-principal bundles.

Clarification of foundational results in equivariant bundle theory.

Proofs of equivariant generalizations of classical non-equivariant results.

Abstract

We compare two notions of $G$-fiber bundles and $G$-principal bundles in the literature, with an aim to clarify early results in equivariant bundle theory that are needed in current work of equivariant algebraic topology. We also give proofs of some equivariant generalizations of well-known non-equivariant results involving the classifying space.