Spherical space forms – Homotopy self-equivalences and homotopy types, the case of the groups Z/a⋊(Z/b×TL2(Fp))

Let G=Z/a⋊μ(Z/b×TL2(Fp)) 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(2dn−1), where 2d is a period of G. Next, the group E(X(2dn−1)/α) of homotopy self-equivalences of spherical space forms X(2dn−1)/α associated with such G-actions α on X(2dn−1) are studied. Similar results for the rest of finite periodic groups have been obtained recently and they are described in the introduction.