Characterizations of $G$-ANR spaces and inverse limits

TL;DR

This paper characterizes when a metrizable $G$-space is a $G$-ANR, establishing conditions involving domination, density, and inverse limits with bonding maps that are fine $G$-homotopy equivalences.

Contribution

It provides new criteria for $G$-ANR spaces in the context of compact groups, especially involving inverse limits and homotopy properties.

Findings

A $G$-space is a $G$-ANR if it dominates a $G$-ANR via a fine $G$-homotopy equivalence.

A $G$-space is a $G$-ANR if it is $G$-homotopy dense in a $G$-ANR.

Inverse limits of $G$-ANR spaces with bonding maps that are fine $G$-homotopy equivalences are $G$-ANR.

Abstract

In this paper we prove that, for a compact group $G$, a metrizable $G$-space is a $G$-ANR under the following asumptions: (1) if it dominates a $G$-ANR space through a fine $G$-homotopy equivalence; (2) if it is $G$-homotopy dense in a $G$-ANR; (3) if it contains a $G$-ANR as a $G$-homotopy dense subset; (4) if it is the inverse limit of an inverse sequence of $G$-ANR spaces with bonding maps that are fine $G$-homotopy equivalences.