The equivariant covering homotopy property

TL;DR

This work extends the equivariant covering homotopy property (ECHP) to generalized equivariant bundles $(\Pi; \Gamma)$ with fibre $F$ and provides a self-contained introduction to the theory. It develops an inductive proof that avoids Bierstone’s condition, showing numerable $(\Pi; \Gamma)$-bundles yield $h\Gamma$-fibrations and thus satisfy the ECHP, while clarifying the link to Hurewicz fibrations. The paper also establishes an equivalence between $(\Pi; \Gamma)$-bundles with fibre $F$ and principal $(\Pi; \Gamma)$-bundles, enabling a unified perspective via associated principal bundles. By situating ECHP within the framework of equivariant fibrations and Eilenberg–MacLane spaces, the results generalize classical cases and illuminate the role of numerability and slices in the equivariant setting.

Abstract

In this paper, we explain how the more general context of generalised equivariant bundles allows for a simple inductive proof of the ECHP. We also make clear the link between the ECHP and the theory of Hurewicz fibrations.

Theorems & Definitions (35)

• Definition 1
• Remark 2
• Definition 3
• Theorem 4
• Definition 5
• Definition 6
• Remark 7
• Remark 8
• Theorem 9
• proof