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# Alexandroff Manifolds and Homogeneous Continua

Published:2013-05-26
Printed: Jun 2014
• A. Karassev,
Department of Computer Science and Mathematics, Nipissing University, North Bay, ON, P1B 8L7
• V. Todorov,
Department of Mathematics, UACG, Sofia, Bulgaria
• V. Valov,
Department of Computer Science and Mathematics, Nipissing University, North Bay, ON, P1B 8L7
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## Abstract

ny homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, 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 homogeneous metric continuum $X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq 1$, cannot be separated by a compactum $K$ with $\check{H}^{n-1}(K;G)=0$ and $\dim_G K\leq n-1$. This provides a partial answer to a question of Kallipoliti-Papasoglu whether any two-dimensional homogeneous Peano continuum cannot be separated by arcs.
 Keywords: Cantor manifold, cohomological dimension, cohomology groups, homogeneous compactum, separator, $V^n$-continuum
 MSC Classifications: 54F45 - Dimension theory [See also 55M10] 54F15 - Continua and generalizations

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