Alexandroff type manifolds and homology manifolds
V. Todorov, V. Valov

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) -manifolds, where 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 -space of dimension is cyclic in dimension : If is a homogeneous metric compactum with , then for every set , which is cutting between two disjoint open subsets of . Another implication of Theorem 3.4 (Corollary 3.6) provides an analog of the classical result of Mazurkiewicz \cite{ma} that no region in can be cut by a subset of dimension . Concerning homology manifolds, it is shown that if is arcwise connected complete metric space which is either a homology -manifold over a group…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology · Topological and Geometric Data Analysis
