Homology manifolds and homogeneous compacta

TL;DR

This paper characterizes when certain homogeneous ANR spaces are almost homology manifolds by providing necessary and sufficient local conditions, advancing understanding of their topological and homological properties.

Contribution

It establishes a precise criterion for identifying almost homology n-manifolds among homogeneous ANR spaces, linking local homological conditions to global topological structure.

Findings

Characterization of almost homology n-manifolds in homogeneous ANR spaces

Necessary and sufficient local conditions for such manifolds

Extension of homology manifold theory to homogeneous spaces

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.