Spherical space forms – Homotopy self-equivalences and homotopy types, the case of the groups

Let G = Z / a ⋊ μ ( Z / b × TL 2 ( F p ) ) and X ( n ) be an n-dimensional CW-complex with the homotopy type of the n-sphere. We determine the automorphism group Aut ( G ) and then compute the number of distinct homotopy types of spherical space forms with respect to free and cellular G-actions on all CW-complexes X ( 2 d n − 1 ) , where 2d is a period of G. Next, the group E ( X ( 2 d n − 1 ) / α ) of homotopy self-equivalences of spherical space forms X ( 2 d n − 1 ) / α associated with such G-actions α on X ( 2 d n − 1 ) are studied. Similar results for the rest of finite periodic groups have been obtained recently and they are described in the introduction.