Bockstein theorem for nilpotent groups
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- by M. Cencelj, J. Dydak, A. Mitra and A. Vavpetič PDF
- Proc. Amer. Math. Soc. 138 (2010), 1501-1510 Request permission
Abstract:
We extend the definition of Bockstein basis $\sigma (G)$ to nilpotent groups $G$. A metrizable space $X$ is called a Bockstein space if $\operatorname {dim}_G(X) = \sup \{\operatorname {dim}_H(X) | H\in \sigma (G)\}$ 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 $\sup \{\operatorname {dim}_H(X) | H\in \sigma (G)\}\leq 1$.
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
- MR Author ID: 819244
- Email: mitra@math.utk.edu, atish.mitra@gmail.com
- A. Vavpetič
- Affiliation: Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani, Jadranska ulica 19, SI-1111 Ljubljana, Slovenija
- Email: ales.vavpetic@fmf.uni-lj.si
- 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
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 1501-1510
- MSC (2010): Primary 54F45; Secondary 55M10, 54C20
- DOI: https://doi.org/10.1090/S0002-9939-09-10143-0
- MathSciNet review: 2578545