Alexandroff Manifolds and Homogeneous Continua

Karassev, A. ; Todorov, V. ; Valov, V.

Canadian mathematical bulletin, Tome 57 (2014) no. 2, pp. 335-343

Résumé

We prove the following result announced by the second and third authors: Any homogeneous, metric $ANR$ -continuum is a $V_{G}^{n}$ -continuum provided ${{\dim}_{G}}X\,=\,n\,\ge \,1$ and ${{\overset{\vee }{\mathop{H}}\,}^{n}}\left( X;\,G \right)\,\ne \,0$ , where $G$ is a principal ideal domain. This implies that any homogeneous $n$ -dimensional metric $ANR$ -continuum is a ${{V}^{n}}$ -continuum in the sense of Alexandroff. We also prove that any finite-dimensional cyclic in dimension $n$ homogeneous metric continuum $X$ , satisfying ${{\overset{\vee }{\mathop{H}}\,}^{n}}\left( X;\,G \right)\,\ne \,0$ for some group $G$ and $n\,\ge \,1$ , cannot be separated by a compactum $K$ with ${{\overset{\vee }{\mathop{H}}\,}^{n-1}}\left( K;\,G \right)\,=\,0$ and ${{\dim}_{G}}K\,\le \,n\,-\,1$ . This provides a partial answer to a question of Kallipoliti–Papasoglu as to whether a two-dimensional homogeneous Peano continuum can be separated by arcs.

Détail

DOI : 10.4153/CMB-2013-010-8

Mots-clés : 54F45, 54F15, Cantor manifold, cohomological dimension, cohomology groups, homogeneous compactum, separator, Vn -continuum

Bibliographie

[1] [1] Alexandroff, P. S., Die Kontinua (V p )–eine Verschärfung der Cantorschen Mannigfaltigkeiten. Monatsh. Math. 61 (1957), 67–76. Google Scholar | DOI

[2] [2] Bing, R. H. and K. Borsuk, Some remarks concerning topologically homogeneous spaces. Ann. of Math. 81 (1965), 100–111. Google Scholar | DOI

[3] [3] Choi, J. S., Properties of n-bubbles in n-dimensional compacta and the existence of (n–1)-bubbles in n-dimensional clcn compacta. Topology Proc. 23 (1998), Spring, 101–120. Google Scholar

[4] [4] Dranishnikov, A. N., On a problem of Aleksandrov P. S.. Mat. Sb. 135 (1988), no. 4, 551–557, 560; translation in Math. USSR-Sb. 63 (1989), no. 2, 539–545. Google Scholar

[5] [5] Dranishnikov, A. N., Universal Menger compacta and universal mappings. (Russian) Mat. Sb. 129 (1986), no. 1, 121–139, 160. Google Scholar

[6] [6] Dydak, J. and Koyama, A., Cohomological dimension of locally connected compacta. Topology Appl. 113 (2001), no. 1–3, 39–50. Google Scholar | DOI

[7] [7] Dydak, J. and Wash, J., Infinite-dimensional compacta having cohomological dimension two: an application of the Sullivan conjecture. Topology 32 (1993), no. 1, 93–104. Google Scholar | DOI

[8] [8] Effros, E. G., Transformation groups and C*-algebras. Ann. of Math. 81 (1965), 38–55. Google Scholar | DOI

[9] [9] Hadžiivanov, N., Strong Cantor manifolds. (Russian) C. R. Acad. Bulgare Sci. 30 (1977), no. 9. 1247–1249. Google Scholar

[10] [10] Hadžiivanov, N. and Todorov, V., On non-Euclidean manifolds. (Russian) C. R. Acad. Bulgare Sci. 33 (1980), no. 4, 449–452. Google Scholar

[11] [11] Kallipoliti, M. and Papasoglu, P., Simply connected homogeneous continua are not separated by arcs. Topology Appl. 154 (2007), no. 17, 3039–3047. Google Scholar | DOI

[12] [12] Karassev, A., Krupski, P., Todorov, V., and Valov, V., Generalized Cantor manifolds and homogeneity. Houston J. Math. 38 (2012), no. 2, 583–609. Google Scholar

[13] [13] Karimov, U. and Repovš, D., On Hn-bubles in n-dimensional compacta. Colloq. Math. 75 (1998), no. 1, 39–51.10.4064/cm-75-1-39-51 Google Scholar | DOI

[14] [14] Krupski, P., Homogeneity and Cantor manifolds. Proc. Amer. Math. Soc. 109 (1990), no. 4, 1135–1142. Google Scholar | DOI

[15] [15] Kuperberg, W., On certain homological properties of finite-dimensional compacta. Carries, minimal carries and bubbles. Fund. Math. 83 (1973), no. 1, 7–23. Google Scholar

[16] [16] Stefanov, S. T., A cohomological analogue of Vn-continua and a theorem of Mazurkiewicz. (Russian) Serdica 12 (1986), no. 1, 88–94. Google Scholar

[17] [17] Todorov, V., Irreducibly cyclic compacta and Cantor manifolds. Proc. of the Tenth Spring Conf. of the Union of Bulg. Math. (Sunny Beach, April 6-9, 1981). Google Scholar

[18] [18] Todorov, V. and Valov, V., Generalized Cantor manifolds and indecomposable continua. Questions Answers Gen. Topology 30 (2012), no. 2, 93–102. Google Scholar

[19] [19] Urysohn, P., Memoire sur les multiplicites cantoriennes. Fund. Math. 7 (1925), 30–137.10.4064/fm-7-1-30-137 Google Scholar | DOI

[20] [20] Yokoi, K., Bubbly continua and homogeneity. Houston J. Math. 29 (2003), no. 2, 337–343. Google Scholar