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