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

 

Semi-free actions of zero-dimensional
compact groups on Menger compacta


Author: Katsuro Sakai
Journal: Proc. Amer. Math. Soc. 125 (1997), 2809-2813
MSC (1991): Primary 54F15, 54H25, 54H15; Secondary 57S10, 22C05
MathSciNet review: 1415368
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Abstract: Let $\mu ^{n}$ be the $n$-dimensional universal Menger compactum, $X$ a $Z$-set in $\mu ^{n}$ and $G$ a metrizable zero-dimensional compact group with $e$ the unit. It is proved that there exists a semi-free $G$-action on $\mu ^{n}$ such that $X$ is the fixed point set of every $g \in G \smallsetminus \{e\}$. As a corollary, it follows that each compactum with $\dim \leqslant n$ can be embedded in $\mu ^{n}$ as the fixed point set of some semi-free $G$-action on $\mu ^{n}$.


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Additional Information

Katsuro Sakai
Affiliation: Institute of Mathematics, University of Tsukuba, Tsukuba-city 305, Japan
Email: sakaiktr@sakura.cc.tsukuba.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04031-8
PII: S 0002-9939(97)04031-8
Keywords: The fixed point set, semi-free action, $0$-dimensional compact group, the $n$-dimensional universal Menger compactum
Received by editor(s): April 16, 1994
Received by editor(s) in revised form: April 28, 1996
Communicated by: James West
Article copyright: © Copyright 1997 American Mathematical Society