Local cohomological properties of homogeneous ANR compacta

V. Valov¹

¹Department of Computer Science and Mathematics Nipissing University 100 College Drive, P.O. Box 5002 North Bay, ON, P1B 8L7, Canada

Fundamenta Mathematicae, Tome 233 (2016) no. 3, pp. 257-270

Résumé

In accordance with the Bing–Borsuk conjecture, we show that if $X$ is an $n$-dimensional homogeneous metric ANR continuum and $x\in X$, then there is a local basis at $x$ consisting of connected open sets $U$ such that the cohomological properties of $\overline U$ and ${\rm bd}\,U$ are similar to the properties of the closed ball $\mathbb B^n\subset \mathbb R^n$ and its boundary $\mathbb S^{n-1}$. We also prove that a metric ANR compactum $X$ of dimension $n$ is dimensionally full-valued if and only if the group $H_n(X,X\setminus x;\mathbb Z)$ is not trivial for some $x\in X$. This implies that every $3$-dimensional homogeneous metric ANR compactum is dimensionally full-valued.

Détail

DOI : 10.4064/fm93-12-2015

Keywords: accordance bing borsuk conjecture n dimensional homogeneous metric anr continuum there local basis consisting connected sets cohomological properties overline similar properties closed ball mathbb subset mathbb its boundary mathbb n prove metric anr compactum dimension dimensionally full valued only group x setminus mathbb trivial implies every dimensional homogeneous metric anr compactum dimensionally full valued