2013/01/13 by Vladimir Todorov, Todorov, V., Vesko Valov +1 · 1 citation
Computer Science · Mathematics · #54F45 #55M10 #Algebraic Topology (math.AT) #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1301.2809
openalex publication_date 2013/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce and investigate the notion of (strong) KnG-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 \citebb, 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 \checkHn(X;G)≠ 0, then \checkHn-1(M;G)≠ 0 for every set M⊂ 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 \citema that no region in \mathbb Rn can be cut by a subset of dimension ≤ 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 Hk(X,X∖ x;G)=0 for every x∈ X and k≤ n-1, where X is a homogeneous locally compact metric ANR.