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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
DOI: https://doi.org/10.1090/proc/13484
Published electronically: February 28, 2018
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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).

References [Enhancements On Off] (What's this?)

  • [1] P. Alexandroff, Introduction to homological dimension theory and general combinatorial topology, Nauka, Moscow, 1975 (in Russian).
  • [2] R. H. Bing and K. Borsuk, Some remarks concerning topologically homogeneous spaces, Ann. of Math. (2) 81 (1965), 100-111. MR 0172255
  • [3] Glen E. Bredon, Sheaf theory, 2nd ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR 1481706
  • [4] Edward G. Effros, Transformation groups and $ C^{\ast} $-algebras, Ann. of Math. (2) 81 (1965), 38-55. MR 0174987
  • [5] William S. Massey, Homology and cohomology theory, An approach based on Alexander-Spanier cochains, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 46, Marcel Dekker, Inc., New York-Basel, 1978. MR 0488016
  • [6] Deane Montgomery, Locally homogeneous spaces, Ann. of Math. (2) 52 (1950), 261-271. MR 0043794
  • [7] E. G. Skljarenko, Homology theory and the exactness axiom, Uspehi Mat. Nauk 24 (1969), no. 5 (149), 87-140 (Russian). MR 0263071
  • [8] N. E. Steenrod, Regular cycles of compact metric spaces, Ann. of Math. (2) 41 (1940), 833-851. MR 0002544
  • [9] V. Todorov and V. Valov, Alexandroff type manifolds and homology manifolds, Houston J. Math. 40 (2014), no. 4, 1325-1346. MR 3298753
  • [10] V. Valov, Homogeneous $ ANR$-spaces and Alexandroff manifolds, Topology Appl. 173 (2014), 227-233. MR 3227218
  • [11] V. Valov, Local cohomological properties of homogeneous ANR compacta, Fund. Math. 233 (2016), no. 3, 257-270. MR 3480120

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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
Email: veskov@nipissingu.ca

DOI: https://doi.org/10.1090/proc/13484
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

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