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

 

Bockstein theorem for nilpotent groups


Authors: M. Cencelj, J. Dydak, A. Mitra and A. Vavpetic
Journal: Proc. Amer. Math. Soc. 138 (2010), 1501-1510
MSC (2010): Primary 54F45; Secondary 55M10, 54C20
Published electronically: November 23, 2009
MathSciNet review: 2578545
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Abstract | References | Similar Articles | Additional Information

Abstract: We extend the definition of Bockstein basis $ \sigma(G)$ to nilpotent groups $ G$. A metrizable space $ X$ is called a Bockstein space if $ \operatorname{dim}_G(X) = \sup\{\operatorname{dim}_H(X) \vert H\in \sigma(G)\}$ for all Abelian groups $ G$. The Bockstein First Theorem says that all compact spaces are Bockstein spaces.

Here are the main results of the paper:

Theorem 0.1. Let $ X$ be a Bockstein space. If $ G$ is nilpotent, then $ \operatorname{dim}_G(X) \leq 1$ if and only if $ \sup\{\operatorname{dim}_H(X) \vert H\in\sigma(G)\}\leq 1$.

Theorem 0.2. $ X$ is a Bockstein space if and only if $ \operatorname{dim}_{{\mathbf{Z}}_{(l)}} (X) = \operatorname{dim}_{\hat{Z}_{(l)}}(X)$ for all subsets $ l$ of prime numbers.


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Additional Information

M. Cencelj
Affiliation: Institute of Mathematics, Physics, and Mechanics, Jadranska ulica 19, SI-1111 Ljubljana, Slovenija
Email: matija.cencelj@guest.arnes.si

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

A. Mitra
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Address at time of publication: University of South Florida, 140 Seventh Avenue South, St. Petersburg, Florida 33701
Email: mitra@math.utk.edu, atish.mitra@gmail.com

A. Vavpetic
Affiliation: Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani, Jadranska ulica 19, SI-1111 Ljubljana, Slovenija
Email: ales.vavpetic@fmf.uni-lj.si

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10143-0
PII: S 0002-9939(09)10143-0
Keywords: Extension dimension, cohomological dimension, absolute extensor, nilpotent groups.
Received by editor(s): September 23, 2008
Received by editor(s) in revised form: April 21, 2009
Published electronically: November 23, 2009
Additional Notes: This work supported in part by the Slovenian-USA research grant BI–US/05-06/002 and the ARRS grants P1-0292-0101 and J1-2057-0101
The second-named author was partially supported by MEC, MTM2006-0825.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2009 American Mathematical Society