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Local homological properties and cyclicity of homogeneous ANR-compacta

Author: V. Valov
Journal: Proc. Amer. Math. Soc. 146 (2018), 2697-2705
MSC (2010): Primary 55M10, 55M15; Secondary 54F45, 54C55
Published electronically: February 28, 2018
MathSciNet review: 3778169
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Abstract: In accordance with the Bing-Borsuk conjecture, we show that if $ X$ is an $ n$-dimensional homogeneous metric $ ANR$-compactum and $ x\in X$, then there is a local basis at $ x$ consisting of connected open sets $ U$ such that the homological properties of $ \overline U$ and $ bd\,\overline 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 discuss also the following questions raised by Bing-Borsuk [Ann. of Math. (2) 81 (1965), 100-111], where $ X$ is a homogeneous $ ANR$-compactum with $ \dim X=n$:

  • Is it true that $ X$ is cyclic in dimension $ n$?
  • Is it true that no non-empty closed subset of $ X$, acyclic in dimension $ n-1$, separates $ X$?
It is shown that both questions simultaneously have positive or negative answers, and a positive solution to each one of them implies a solution to another question of Bing-Borsuk (whether every finite-dimensional homogenous metric $ AR$-compactum is a point).

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Additional Information

V. Valov
Affiliation: Department of Computer Science and Mathematics, Nipissing University, 100 College Drive, P.O. Box 5002, North Bay, Ontario, P1B 8L7, Canada

Keywords: Bing-Borsuk conjecture for homogeneous compacta, dimensionally full-valued compacta, homology membrane, homological dimension, homology groups, homogeneous metric $ANR$-compacta
Received by editor(s): January 25, 2016
Received by editor(s) in revised form: September 18, 2016
Published electronically: February 28, 2018
Additional Notes: The author was partially supported by NSERC Grant 261914-13.
Communicated by: Kevin Whyte
Article copyright: © Copyright 2018 American Mathematical Society