Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group (Z/a ⋊ Z/b) × SL 2 ( p )

Golasiński, Marek ; Gonçalves, Daciberg Lima

Canadian mathematical bulletin, Tome 50 (2007) no. 2, pp. 206-214

Résumé

Let $G=\left( \mathbb{Z}/a\rtimes \mathbb{Z}/b \right)\times \text{S}{{\text{L}}_{2}}\left( {{\mathbb{F}}_{p}} \right)$ , and let $X\left( n \right)$ be an $n$ -dimensional $CW$ -complex of the homotopy type of an $n$ -sphere. We study the automorphism group $\text{Aut}\left( G \right)$ 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 $X\left( 2dn-1 \right)$ , where $2d$ is the period of $G$ . The groups $\varepsilon \left( X\left( 2dn-1 \right)/\mu\right)$ of self homotopy equivalences of space forms $X\left( 2dn-1 \right)/\mu$ associated with free and cellular $G$ -actions $\mu$ on $X\left( 2dn-1 \right)$ are determined as well.

Détail

DOI : 10.4153/CMB-2007-022-5

Mots-clés : 55M35, 55P15, 20E22, 20F28, 57S17, 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