Spherical space forms -- homotopy self-equivalences and homotopy types, the case of the groups \(\mathbb Z/a\rtimes (\mathbb Z/b \times TL_2(\mathbb F_p))\)
type: Item
instance of: scholarly article
title: Spherical space forms -- homotopy self-equivalences and homotopy types, the case of the groups \(\mathbb Z/a\rtimes (\mathbb Z/b \times TL_2(\mathbb F_p))\)
zbMATH Open document ID: 1185.55004
author: Marek Golasiński
author: Daciberg Lima Gonçalves
published in: Topology and its Applications
publication date: 2009-11-05T00:00:00Z
review text: Let \(X(n)\) be an \(n\)-dimensional CW complex with the homotopy type of an \(n\)-sphere, and let \(\Aut(G)\) be the automorphism group of a group \(G\). Recall that the cohomology of a discrete group \(G\) acting freely and cellularly on \(X(n)\) is periodic by using the result due to Swan, and it is known that such a group belongs to one of the families of the groups I to VI in the Suzuki-Zassenhaus classification. It is also known that the set of homotopy types of spherical space forms of all free and cellularly \(G\)-actions on \(X(2n-1)\) is in one-to-one correspondence with the set of the orbits which contain a generator of \(H^{2n}(G)\cong \mathbb Z/| G|\) under the action of \(\pm \text{Aut}(G)\). In this paper, the authors consider the group \(G=\mathbb Z /a \rtimes (\mathbb Z \times Tl_2(\mathbb F_p))\) of the family VI. In particular, for this group \(G\) they determine the group \(\Aut(G)\) and compute the number of distinct homotopy types of spherical space forms of \(X(2dn-1)\), where \(2d\) is a period. Moreover, they study the group \({\mathcal E}(X)\) of homotopy self-equivalences of \(X\) for the orbit space \(X=X(2dn-1)/G\).
reviewed by: Kohhei Yamaguchi
Mathematics Subject Classification ID: 55M35
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 5625472
zbMATH Keywords: automorphism group
zbMATH Keywords: free and cellular \(G\)-action
zbMATH Keywords: group of self-homotopy equivalences
zbMATH Keywords: Lyndon-Hochschild-Serre space form
MaRDI profile type: Publication
full work available at URL: j.topol.2009.08.004
OpenAlex ID: W2062479463
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DOI: 10.1016/J.TOPOL.2009.08.004
description: scientific article; zbMATH DE number 5625472