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Characterizations of $G$-ANR spaces and inverse limits

Sergey A. Antonyan, Aura Lucina Kantún-Montiel, Jesús Eduardo Mata-Cano, Armando Mata-Romero

TL;DR

The paper addresses when metrizable $G$-spaces are $G$-ANR under conditions such as domination by a fine $G$-homotopy equivalence, $G$-homotopy density, containing a $G$-ANR as a dense subset, and inverse limits with bonding maps that are fine $G$-homotopy equivalences. It develops $G$-$\mathcal{U}$-homotopies and refinement tools to control equivariant retract properties and proves a Kozlowski-type equivariant theorem. It also extends Curtis’s inverse-limit result to the equivariant setting, showing that the inverse limit of completely metrizable $G$-ANRs with fine bonding maps is again a $G$-ANR (and a $G$-AR when appropriate). Overall, the work provides a cohesive framework for constructing and recognizing $G$-ANRs via domination, density, and inverse-limit constructions, with explicit extension techniques for invariant maps.

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.

Characterizations of $G$-ANR spaces and inverse limits

TL;DR

The paper addresses when metrizable -spaces are -ANR under conditions such as domination by a fine -homotopy equivalence, -homotopy density, containing a -ANR as a dense subset, and inverse limits with bonding maps that are fine -homotopy equivalences. It develops --homotopies and refinement tools to control equivariant retract properties and proves a Kozlowski-type equivariant theorem. It also extends Curtis’s inverse-limit result to the equivariant setting, showing that the inverse limit of completely metrizable -ANRs with fine bonding maps is again a -ANR (and a -AR when appropriate). Overall, the work provides a cohesive framework for constructing and recognizing -ANRs via domination, density, and inverse-limit constructions, with explicit extension techniques for invariant maps.

Abstract

In this paper we prove that, for a compact group , a metrizable -space is a -ANR under the following asumptions: (1) if it dominates a -ANR space through a fine -homotopy equivalence; (2) if it is -homotopy dense in a -ANR; (3) if it contains a -ANR as a -homotopy dense subset; (4) if it is the inverse limit of an inverse sequence of -ANR spaces with bonding maps that are fine -homotopy equivalences.
Paper Structure (6 sections, 14 theorems, 37 equations)

This paper contains 6 sections, 14 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Basic notations
  3. $G$-$\mathcal{U}$-homotopies
  4. $G$-ANR spaces and fine $G$-homotopy equivalences
  5. $G$-homotopy dense subsets and $G$-ANR spaces
  6. Equivariant inverse sequences and $G$-ANR spaces

Key Result

Lemma 3.1

Let $f:X\to Y$ be a $G$-map such that $f(X)$ is dense in $Y$. Let $\mathcal{U}$ be an open cover of $Y$ and $\varphi: Y\to X$ be a $G$-map such that $\varphi f$ is $f^{-1}(\mathcal{U})$-close to the identity map $id_{X}$. Then $f\varphi$ is $\text{St}(\mathcal{U})$-close to $id_{Y}$.

Theorems & Definitions (22)

  • Lemma 3.1
  • proof
  • Proposition 3.2: Antonyan2005
  • Theorem 3.3: Ale2007
  • Theorem 3.4: Antonyan1988
  • Corollary 3.5
  • Theorem 4.1: Ale2007
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • ...and 12 more