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

Authors :
Golasiński, Marek
Gonçalves, Daciberg Lima
Jimenez, Rolando
Source :
Proceedings of the Edinburgh Mathematical Society; May2018, Vol. 61 Issue 2, p305-327, 23p
Publication Year :
2018

Abstract

Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional C W-complex Σ(2 n) which has the homotopy type of the 2 n -sphere 𝕊<superscript>2 n </superscript>. Then, that action induces a homomorphism G → Aut(H <superscript>2 n </superscript>(Σ(2 n))). 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(2 n). Those groups consist of free groups and semi-direct products F ⋊ ℤ<subscript>2</subscript> 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(2 n) with dim Σ(2 n) ≤ 2 n + 1. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
00130915
Volume :
61
Issue :
2
Database :
Complementary Index
Journal :
Proceedings of the Edinburgh Mathematical Society
Publication Type :
Academic Journal
Accession number :
136611881
Full Text :
https://doi.org/10.1017/S0013091517000207