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