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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hereditarily aspherical compacta


Authors: Jerzy Dydak and Katsuya Yokoi
Journal: Proc. Amer. Math. Soc. 124 (1996), 1933-1940
MSC (1991): Primary 55M10, 54F45
MathSciNet review: 1307513
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Abstract: The notion of (strongly) hereditarily aspherical compacta introduced by Daverman (1991) is modified. The main results are:

Theorem. If $X\in LC^{1}$ is a hereditarily aspherical compactum, then $X\in $ ANR. In particular, $X$ is strongly hereditarily aspherical.

Theorem. Suppose $f:X\to Y$ is a cell-like map of compacta and $f^{-1}(A)$ is shape aspherical for each closed subset $A$ of $Y$. Then

1.
Y is hereditarily shape aspherical,
2.
$f$ is a hereditary shape equivalence,
3.
$\dim X\ge \dim Y$.

Theorem. Suppose $G$ is a group containing integers. Then the following conditions are equivalent:

1.
$\dim X\le 2$ and $\dim _{G}X=1$,
2.
$\dim _{G*_{{\mathbf Z} }G}X=1$.

Theorem. Suppose $G$ is a group containing integers. If $\dim X\le 2$ and $\dim _{G}X=1$, then $X$ is hereditarily shape aspherical.

Theorem. Let $X$ be a two-dimensional, locally connected and semilocally simply connected compactum. Then, for any compactum $Y$

\begin{equation*}\dim (X \times Y) = \dim X + \dim Y.\end{equation*}


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Additional Information

Jerzy Dydak
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Email: dydak@math.utk.edu

Katsuya Yokoi
Affiliation: Institute of Mathematics, University of Tsukuba, Tsukuba-shi, Ibaraki, 305, Japan
Email: yokoi@sakura.cc.tsukuba.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03221-2
PII: S 0002-9939(96)03221-2
Keywords: Dimension, cohomological dimension, aspherical compacta, ANR's, absolute extensors, cell-like maps
Received by editor(s): April 6, 1994
Received by editor(s) in revised form: November 19, 1994
Communicated by: James West
Article copyright: © Copyright 1996 American Mathematical Society