Alexandroff type manifolds and homology manifolds

TL;DR

This paper introduces new concepts of $K^n_G$-manifolds and explores their properties, providing partial solutions to the Bing-Borsuk problem and extending classical results about cuts in Euclidean spaces, with implications for homology manifolds.

Contribution

It defines (strong) $K^n_G$-manifolds, proves their properties, and applies these to partial solutions of the Bing-Borsuk problem and classical cutting theorems, extending the theory of homology manifolds.

Findings

Partial answer to Bing-Borsuk problem for homogeneous $ANR$-spaces.

No subset of dimension ≤ n-2 can cut a region in $ eal^n$.

Homology $n$-manifolds or products of spaces are Mazurkiewicz arc $n$-manifolds.

Abstract

We introduce and investigate the notion of (strong) $K^n_G$-manifolds, where $G$ is an abelian group. One of the result related to that notion (Theorem 3.4) implies the following partial answer to the Bing-Borsuk problem \cite{bb}, whether any partition of a homogeneous metric $ANR$-space $X$ of dimension $n$ is cyclic in dimension $n-1$: If $X$ is a homogeneous metric $ANR$ compactum with $\check{H}^{n}(X;G)\neq 0$, then $\check{H}^{n-1}(M;G)\neq 0$ for every set $M\subset X$, which is cutting $X$ between two disjoint open subsets of $X$. Another implication of Theorem 3.4 (Corollary 3.6) provides an analog of the classical result of Mazurkiewicz \cite{ma} that no region in $\mathbb R^n$ can be cut by a subset of dimension $\leq n-2$. Concerning homology manifolds, it is shown that if $X$ is arcwise connected complete metric space which is either a homology $n$-manifold over a group $G$ or a product of at least $n$ metric spaces, then $X$ is a Mazurkiewicz arc $n$-manifold. We also introduce a property which guarantees that $H_k(X,X\setminus x;G)=0$ for every $x\in X$ and $k\leq n-1$, where $X$ is a homogeneous locally compact metric $ANR$.