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Every CW-complex is a classifying space for proper bundles

Every CW-complex is a classifying space for proper bundles
Every CW-complex is a classifying space for proper bundles
We prove that, up to homotopy equivalence, every connected CW-complex is the quotient of a contractible complex by a proper action of a discrete group, and that every CW-complex is the quotient of an aspherical complex by an action of a group of order two. These results may be viewed as analogues of the Kan-Thurston theorem, in which the universal free G-space has been replaced by the universal proper G-space. The fact that our result concerns homotopy type (whereas the Kan-Thurston theorem concerns homology) is a reflection of the existence of 'contractible groups', i.e., groups for which the quotient of the universal proper G-space by G is contractible.
CW-complex, homotopy type, actions of discrete groups
0040-9383
539-550
Leary, Ian J.
57bd5c53-cd99-41f9-b02a-4a512d45150e
Nucinkis, Brita E.A.
0b1c337c-36ae-4ef3-add4-b49a7c23810c
Leary, Ian J.
57bd5c53-cd99-41f9-b02a-4a512d45150e
Nucinkis, Brita E.A.
0b1c337c-36ae-4ef3-add4-b49a7c23810c

Leary, Ian J. and Nucinkis, Brita E.A. (2001) Every CW-complex is a classifying space for proper bundles. Topology, 40 (3), 539-550. (doi:10.1016/S0040-9383(99)00073-7).

Record type: Article

Abstract

We prove that, up to homotopy equivalence, every connected CW-complex is the quotient of a contractible complex by a proper action of a discrete group, and that every CW-complex is the quotient of an aspherical complex by an action of a group of order two. These results may be viewed as analogues of the Kan-Thurston theorem, in which the universal free G-space has been replaced by the universal proper G-space. The fact that our result concerns homotopy type (whereas the Kan-Thurston theorem concerns homology) is a reflection of the existence of 'contractible groups', i.e., groups for which the quotient of the universal proper G-space by G is contractible.

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More information

Published date: 2001
Keywords: CW-complex, homotopy type, actions of discrete groups

Identifiers

Local EPrints ID: 29834
URI: http://eprints.soton.ac.uk/id/eprint/29834
ISSN: 0040-9383
PURE UUID: b6e3ea5c-c4f6-413e-bc55-ff8201a21b60
ORCID for Ian J. Leary: ORCID iD orcid.org/0000-0001-8300-4979

Catalogue record

Date deposited: 11 May 2006
Last modified: 16 Mar 2024 04:04

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Contributors

Author: Ian J. Leary ORCID iD
Author: Brita E.A. Nucinkis

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