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Homology manifolds and homogeneous compacta

Vesko Valov

TL;DR

This work characterizes when locally compact homogeneous $ANR$-spaces (and strongly locally homogeneous $ANR$-spaces) are almost homology $n$-manifolds. It introduces the density condition $B(n-1)$ on maps from $\mathbb B^{n-1}$ to $X$ and proves it is both necessary and sufficient for the almost homology $n$-manifold property, via a $Z_{n-1}$-set and homological $Z_n$-set framework and Krupski-type arguments. The results yield that for strongly locally homogeneous $LC^{n-1}$-spaces, the $B(n-1)$ condition implies $X$ is an almost homology $n$-manifold (and conversely), with the key consequence that $\dim X\ge n$ and various corollaries about nowhere locally compact spaces. Overall, the paper advances the understanding of when homogeneous, locally contractible spaces approximate classical manifolds in homological terms, contributing to longstanding questions related to Bing–Borsuk-type conjectures.

Abstract

A non-trivial separable metric space $X$ is called an almost homology $n$-manifold if the homology groups $H_k(X,X\backslash\{x\},\mathbb Z)$ are trivial for all $x\in X$ and all $k=0,1,..,n-1$. We provide a necessary and sufficient condition locally compact homogeneous $ANR$-spaces or strongly locally homogeneous $ANR$-spaces to be almost homology $n$-manifolds.

Homology manifolds and homogeneous compacta

TL;DR

This work characterizes when locally compact homogeneous -spaces (and strongly locally homogeneous -spaces) are almost homology -manifolds. It introduces the density condition on maps from to and proves it is both necessary and sufficient for the almost homology -manifold property, via a -set and homological -set framework and Krupski-type arguments. The results yield that for strongly locally homogeneous -spaces, the condition implies is an almost homology -manifold (and conversely), with the key consequence that and various corollaries about nowhere locally compact spaces. Overall, the paper advances the understanding of when homogeneous, locally contractible spaces approximate classical manifolds in homological terms, contributing to longstanding questions related to Bing–Borsuk-type conjectures.

Abstract

A non-trivial separable metric space is called an almost homology -manifold if the homology groups are trivial for all and all . We provide a necessary and sufficient condition locally compact homogeneous -spaces or strongly locally homogeneous -spaces to be almost homology -manifolds.
Paper Structure (3 sections, 11 theorems, 3 equations)

This paper contains 3 sections, 11 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Prof of Theorem 1.1
  3. Proof of Theorem 1.2

Key Result

Theorem 1.1

Let $X$ be a locally compact homogeneous $ANR$-space. Then $X$ is an almost homology $n$-manifold if and only if $X$ satisfies the following condition $B(n-1)$: the set of all maps $f:\mathbb B^{n-1}\to X$ such that $f(\mathbb B^{n-1})$ has an empty interior is dense in the function space $C(\mathbb

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Proposition 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • Claim 1
  • ...and 9 more