Local structure of homogeneous 𝐴𝑁𝑅-spaces

Valov, Vesko

doi:10.1090/tran/9296

Local structure of homogeneous $ANR$-spaces
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by Vesko Valov;

Trans. Amer. Math. Soc. 378 (2025), 1449-1463

DOI: https://doi.org/10.1090/tran/9296

Published electronically: October 17, 2024

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Abstract:

We investigate to what extend finite-dimensional homogeneous locally compact $ANR$-spaces have common properties with topological manifolds. Specially, the local structure of homogeneous $ANR$-spaces is described. Using that description, we provide a positive solution of the problem whether every finite-dimensional homogeneous metric $ANR$-compactum $X$ is dimensionally full-valued, i.e. $\dim X\times Y=\dim X+\dim Y$ for any metric compactum $Y$.

References

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
Karol Borsuk, Theory of retracts, Monografie Matematyczne [Mathematical Monographs], Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR 216473
J. Bryant, Reflections on the Bing-Borsuk conjecture, Abstract of the talks presented at the 19th Annual Workshop in Geometric Topology, June 13-15, 2002, 2-3.
J. Bryant and S. Ferry, An alpha approximation theorem for homology manifolds: counterexamples to the Bing-Borsuk conjecture, preprint.
Manuel Cárdenas, Francisco F. Lasheras, Antonio Quintero, and Dušan Repovš, On manifolds with nonhomogeneous factors, Cent. Eur. J. Math. 10 (2012), no. 3, 857–862. MR 2902218, DOI 10.2478/s11533-012-0026-6
J. S. Choi, Properties of $n$-bubbles in $n$-dimensional compacta and the existence of $(n-1)$-bubbles in $n$-dimensional $\textrm {clc}^n$ compacta, Proceedings of the 1998 Topology and Dynamics Conference (Fairfax, VA), 1998, pp. 101–120. MR 1743802
A. Dranishnikov, Cohomological dimension theory of compact metric spaces, Topology Atlas invited contributions, http://at.yorku.ca/topology.taic.html (see also arXiv:math/0501523).
A. N. Dranishnikov, On a problem of P. S. Aleksandrov, Mat. Sb. (N.S.) 135(177) (1988), no. 4, 551–557, 560 (Russian); English transl., Math. USSR-Sb. 63 (1989), no. 2, 539–545. MR 942139, DOI 10.1070/SM1989v063n02ABEH003290
A. N. Dranishnikov, Homological dimension theory, Uspekhi Mat. Nauk 43 (1988), no. 4(262), 11–55, 255 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 4, 11–63. MR 969565, DOI 10.1070/RM1988v043n04ABEH001900
Jerzy Dydak, Cohomological dimension and metrizable spaces, Trans. Amer. Math. Soc. 337 (1993), no. 1, 219–234. MR 1153013, DOI 10.1090/S0002-9947-1993-1153013-X
Edward G. Effros, Transformation groups and $C^{\ast }$-algebras, Ann. of Math. (2) 81 (1965), 38–55. MR 174987, DOI 10.2307/1970381
Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, NJ, 1952. MR 50886, DOI 10.1515/9781400877492
V. V. Fedorchuk, On homogeneous Pontryagin surfaces, Dokl. Akad. Nauk 404 (2005), no. 5, 601–603 (Russian). MR 2256818
W. Jakobsche, The Bing-Borsuk conjecture is stronger than the Poincaré conjecture, Fund. Math. 106 (1980), no. 2, 127–134. MR 580590, DOI 10.4064/fm-106-2-127-134
Denise M. Halverson and Dušan Repovš, The Bing-Borsuk and the Busemann conjectures, Math. Commun. 13 (2008), no. 2, 163–184. MR 2488667
A. È. Harlap, Local homology and cohomology, homological dimension, and generalized manifolds, Mat. Sb. (N.S.) 96(138) (1975), 347–373, 503 (Russian). MR 645364
Peter J. Huber, Homotopical cohomology and Čech cohomology, Math. Ann. 144 (1961), 73–76. MR 133821, DOI 10.1007/BF01396544
Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, NJ, 1941. MR 6493
Yukihiro Kodama, Note on an absolute neighborhood extensor for metric spaces, J. Math. Soc. Japan 8 (1956), 206–215. MR 81456, DOI 10.2969/jmsj/00830206
Yukihiro Kodama, On a problem of Alexandroff concerning the dimension of product spaces. II, J. Math. Soc. Japan 11 (1959), 94–111. MR 116307, DOI 10.2969/jmsj/01120094
V. Kuz′minov, On $V^{n}$ continua, Dokl. Akad. Nauk SSSR 139 (1961), 24–27 (Russian). MR 165523
V. Kuz’minov, Homological dimension theory, Russian Math. Surveys 23 (1968), no. 1, 1–45.
J. M. Łysko, Some theorems concerning finite dimensional homogeneous ANR-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), no. 7, 491–496 (English, with Russian summary). MR 413110
Deane Montgomery, Locally homogeneous spaces, Ann. of Math. (2) 52 (1950), 261–271. MR 43794, DOI 10.2307/1969469
Hans-Peter Seidel, Locally homogeneous ANR-spaces, Arch. Math. (Basel) 44 (1985), no. 1, 79–81. MR 778995, DOI 10.1007/BF01193784
E. G. Skljarenko, Homology theory and the exactness axiom, Uspehi Mat. Nauk 24 (1969), no. 5(149), 87–140 (Russian). MR 263071
Akira Tominaga and Tadashi Tanaka, Convexification of locally connected generalized continua, J. Sci. Hiroshima Univ. Ser. A 19 (1955), 301–306. MR 78677
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
V. Valov, Homological dimension and homogeneous ANR spaces, Topology Appl. 221 (2017), 38–50. MR 3624443, DOI 10.1016/j.topol.2017.02.063
Vesko Valov, Separation of homogeneous connected locally compact spaces, Proc. Amer. Math. Soc. Ser. B 11 (2024), 36–46. MR 4713118, DOI 10.1090/bproc/207

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Local structure of homogeneous $ANR$-spaces
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Abstract: