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# Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$

Published:2007-06-01
Printed: Jun 2007
• Marek Golasiński
• Daciberg Lima Gonçalves
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## Abstract

Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times \SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional $CW$-complex of the homotopy type of an $n$-sphere. We study the automorphism group $\Aut (G)$ in order to compute the number of distinct homotopy types of spherical space forms with respect to free and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$ is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as well.
 Keywords: automorphism group, $CW$-complex, free and cellular $G$-action, group of self homotopy equivalences, Lyndon--Hochschild--Serre spectral sequence, special (linear) group, spherical space form
 MSC Classifications: 55M35 - Finite groups of transformations (including Smith theory) [See also 57S17] 55P15 - Classification of homotopy type 20E22 - Extensions, wreath products, and other compositions [See also 20J05] 20F28 - Automorphism groups of groups [See also 20E36] 57S17 - Finite transformation groups

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