Local homological properties and cyclicity of homogeneous ANR-compacta
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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
- P. Alexandroff, Introduction to homological dimension theory and general combinatorial topology, Nauka, Moscow, 1975 (in Russian).
- R. H. Bing and K. Borsuk, Some remarks concerning topologically homogeneous spaces, Ann. of Math. (2) 81 (1965), 100–111. MR 172255, DOI 10.2307/1970385
- Glen E. Bredon, Sheaf theory, 2nd ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR 1481706, DOI 10.1007/978-1-4612-0647-7
- Edward G. Effros, Transformation groups and $C^{\ast }$-algebras, Ann. of Math. (2) 81 (1965), 38–55. MR 174987, DOI 10.2307/1970381
- William S. Massey, Homology and cohomology theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 46, Marcel Dekker, Inc., New York-Basel, 1978. An approach based on Alexander-Spanier cochains. MR 0488016
- Deane Montgomery, Locally homogeneous spaces, Ann. of Math. (2) 52 (1950), 261–271. MR 43794, DOI 10.2307/1969469
- E. G. Skljarenko, Homology theory and the exactness axiom, Uspehi Mat. Nauk 24 (1969), no. 5 (149), 87–140 (Russian). MR 0263071
- N. E. Steenrod, Regular cycles of compact metric spaces, Ann. of Math. (2) 41 (1940), 833–851. MR 2544, DOI 10.2307/1968863
- V. Todorov and V. Valov, Alexandroff type manifolds and homology manifolds, Houston J. Math. 40 (2014), no. 4, 1325–1346. MR 3298753
- V. Valov, Homogeneous $ANR$-spaces and Alexandroff manifolds, Topology Appl. 173 (2014), 227–233. MR 3227218, DOI 10.1016/j.topol.2014.06.001
- V. Valov, Local cohomological properties of homogeneous ANR compacta, Fund. Math. 233 (2016), no. 3, 257–270. MR 3480120, DOI 10.4064/fm93-12-2015
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
- MR Author ID: 176775
- Email: veskov@nipissingu.ca
- 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
- © Copyright 2018 American Mathematical Society
- 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
- MathSciNet review: 3778169