Alexandroff type manifolds and homology manifolds

Abstract

We introduce and investigate the notion of (strong) $K_G^n$-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 ${\check{H}}^n(X;G)\ne 0$, then ${\check{H}}^{n-1}(M;G)\ne 0$ 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 $\le n-2$. 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 $k\le n-1$, where $X$ is a homogeneous locally compact metric $ANR$.

22 pages

References in corpus (3)

• Cohomological dimension theory of compact metric spaces
• Generalized Cantor manifolds and indecomposable continua
• Mazurkiewicz manifolds and homogeneity

Cited by in corpus (1)

• Homogeneous metric ANR-compacta