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Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group (ℤ/a ⋊ ℤ/b) × SL 2 (
p )
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- Journal:
- Canadian Mathematical Bulletin / Volume 50 / Issue 2 / 01 June 2007
- Published online by Cambridge University Press:
- 20 November 2018, pp. 206-214
- Print publication:
- 01 June 2007
-
- Article
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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.
