Homotopy spherical space forms---a numerical bound for homotopy types

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

“…In particular, for 2d 4 in the light of [14] those groups act on a (4n − 1)-dimensional CW-complex X(4n − 1) for any n 1. Then Corollary 2.7 states that the number of such homotopy types of space forms in case of Z/a Z/b-actions coincides with that number of (4n − 1)-lens spaces studied in [6] or (4n − 1)-spaces forms for a generalized quaternion group developed in [7] …”

“…Taking into account [12] the case of spherical space forms presents a special interest. For lens spaces that group has been first approached in [10] then recovered in [12] and explicitly described in [6,7]. The paper [7] concerns also with those groups of self homotopy equivalences for space forms given by the generalized quaternion group actions.…”

“…Swan [14] has shown that any finite group with periodic cohomology of period 2d acts freely and cellularly on a (2d − 1)-dimensional CW-complex of the homotopy type of a (2d − 1)-sphere. By means of results in [14], it is shown in [15] that the set of homotopy types of spherical space forms of all free cellular G-actions on (2n − 1)-dimensional CW-complexes with the homotopy type of a (2n − 1)-sphere is in one-one correspondence with the orbits, which contain a generator, of H 2n (G) = Z/|G| under the action of ± Aut(G) (see also [6] for another approach). The papers [6,7] deal with the homotopy types of lens spaces and spherical space forms for generalized quaternion groups Q 4m and explicit formulae for numbers of homotopy types of those spherical space forms have been developed.…”

“…By means of results in [14], it is shown in [15] that the set of homotopy types of spherical space forms of all free cellular G-actions on (2n − 1)-dimensional CW-complexes with the homotopy type of a (2n − 1)-sphere is in one-one correspondence with the orbits, which contain a generator, of H 2n (G) = Z/|G| under the action of ± Aut(G) (see also [6] for another approach). The papers [6,7] deal with the homotopy types of lens spaces and spherical space forms for generalized quaternion groups Q 4m and explicit formulae for numbers of homotopy types of those spherical space forms have been developed. There are other finite periodic groups and the Suzuki-Zassenhaus classification of all such groups is presented, e.g., in the table [1, Chapter IV, Theorem 6.15].…”

“…Obviously cyclic groups are included in the family I and generalized quaternion groups in the family II. However results in [6,7] are essential to make crucial calculations and develop main results stated in Theorems 2.6, 3.2 and 3.4.…”