Spherical space forms—Homotopy types and self-equivalences for the groups Z/a⋊Z/b and Z/a⋊(Z/b×Q2i)

Let G be a finite group one of the forms given in the title with period 2 d and X ( n ) an n -dimensional CW-complex with the homotopy type of an n -sphere. We study the automorphism group Aut( G ) to compute the number of distinct homotopy types of spherical space forms with respect to G -actions on all CW-complexes X (2 dn −1). At the end, the groups E (X(2dn−1)/γ) of self homotopy equivalences of space forms X (2 dn −1)/ γ associated with free and cellular G -actions γ on X (2 dn −1) are determined.