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Free and Properly Discontinuous Actions of Groups on Homotopy 2n-spheres

Part of: PL-topology Connections with homological algebra and category theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  08 February 2018

Marek Golasiński ,
Daciberg Lima Gonçalves  and
Rolando Jimenez
Marek Golasiński*
Affiliation:
Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Stoneczna 54 Street, 10-710 Olsztyn, Poland (marekg@matman.uwm.edu.pl)
Daciberg Lima Gonçalves
Affiliation:
Department of Mathematics-IME, University of São Paulo, Rua do Matão 1010, CEP:055080-090, São Paulo, Brasil (dlgoncal@ime.usp.br)
Rolando Jimenez
Affiliation:
Instituto de Matemáticas, Unidad Oaxaca, Universidad Nacional Autónoma de México, Oaxaca, Oax.Mexico (rolando@matcuer.unam.mx)
*
*Corresponding author.
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Abstract

Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional CW-complex Σ(2n) which has the homotopy type of the 2n-sphere 𝕊2n. Then, that action induces a homomorphism G → Aut(H2n(Σ(2n))). We classify all pairs (G, φ), where G is a virtually cyclic group and φ: G → Aut(ℤ) is a homomorphism, which are realizable in the way above and the homotopy types of all possible orbit spaces as well. Next, we consider the family of all groups which have virtual cohomological dimension one and which act on some Σ(2n). Those groups consist of free groups and semi-direct products F ⋊ ℤ2 with F a free group. For a group G from the family above and a homomorphism φ: G → Aut(ℤ), we present an algebraic criterion equivalent to the realizability of the pair (G, φ). It turns out that any realizable pair can be realized on some Σ(2n) with dim Σ(2n) ≤ 2n + 1.

Keywords

homotopy sphereorbit spacecohomological dimensionproperly discontinuous and cellular actionvirtual cohomological dimensionvirtually cyclic group

MSC classification

Primary: 57S30: Discontinuous groups of transformations
Secondary: 20F50: Periodic groups; locally finite groups 20J06: Cohomology of groups 57Q91: Equivariant PL-topology
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 2 , May 2018 , pp. 305 - 327
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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