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

 

Proper actions on topological groups: Applications to quotient spaces


Author: Sergey A. Antonyan
Journal: Proc. Amer. Math. Soc. 138 (2010), 3707-3716
MSC (2010): Primary 22A05, 22F05, 54H11, 54H15, 54F45
Published electronically: May 27, 2010
MathSciNet review: 2661569
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Abstract: Let $ X$ be a Hausdorff topological group and $ G$ a locally compact subgroup of $ X$. We show that the natural action of $ G$ on $ X$ is proper in the sense of R. Palais. This is applied to prove that there exists a closed set $ F\subset X$ such that $ FG=X$ and the restriction of the quotient projection $ X\to X/G$ to $ F$ is a perfect map $ F\to X/G$. This is a key result to prove that many topological properties (among them, paracompactness and normality) are transferred from $ X$ to $ X/G$, and some others are transferred from $ X/G$ to $ X$. Yet another application leads to the inequality $ {dim} X\le {\rm dim} X/G + {dim} G$ for every paracompact topological group $ X$ and a locally compact subgroup $ G$ of $ X$ having a compact group of connected components.


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Additional Information

Sergey A. Antonyan
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autó- noma de México, 04510 México Distrito Federal, México
Email: antonyan@unam.mx

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10504-X
PII: S 0002-9939(2010)10504-X
Keywords: Proper $G$-space, orbit space, locally compact group, dimension.
Received by editor(s): May 15, 2009
Published electronically: May 27, 2010
Additional Notes: The author was supported in part by grants #IN102608 from PAPIIT (UNAM) and #79536 from CONACYT (Mexico)
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2010 American Mathematical Society