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Local cohomological properties of homogeneous ANR compacta

Volume 233 / 2016

V. Valov Fundamenta Mathematicae 233 (2016), 257-270 MSC: Primary 55M10, 55M15; Secondary 54F45, 54C55. DOI: 10.4064/fm93-12-2015 Published online: 16 December 2015

Abstract

In accordance with the Bing–Borsuk conjecture, we show that if $X$ is an $n$-dimensional homogeneous metric ANR continuum and $x\in X$, then there is a local basis at $x$ consisting of connected open sets $U$ such that the cohomological properties of $\overline U$ and ${\rm bd}\,U$ are similar to the properties of the closed ball $\mathbb B^n\subset \mathbb R^n$ and its boundary $\mathbb S^{n-1}$. We also prove that a metric ANR compactum $X$ of dimension $n$ is dimensionally full-valued if and only if the group $H_n(X,X\setminus x;\mathbb Z)$ is not trivial for some $x\in X$. This implies that every $3$-dimensional homogeneous metric ANR compactum is dimensionally full-valued.

Authors

  • V. ValovDepartment of Computer Science and Mathematics
    Nipissing University
    100 College Drive, P.O. Box 5002
    North Bay, ON, P1B 8L7, Canada
    e-mail

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