Spherical space forms -- homotopy self-equivalences and homotopy types, the case of the groups \(\mathbb Z/a\rtimes (\mathbb Z/b \times TL_2(\mathbb F_p))\)
Let \(X(n)\) be an \(n\)-dimensional CW complex with the homotopy type of an \(n\)-sphere, and let \(\Aut(G)\) be the automorphism group of a group \(G\). Recall that the cohomology of a discrete group \(G\) acting freely and cellularly on \(X(n)\) is periodic by using the result due to Swan, and it is known that such a group belongs to one of the families of the groups I to VI in the Suzuki-Zassenhaus classification. It is also known that the set of homotopy types of spherical space forms of all free and cellularly \(G\)-actions on \(X(2n-1)\) is in one-to-one correspondence with the set of the orbits which contain a generator of \(H^{2n}(G)\cong \mathbb Z/| G|\) under the action of \(\pm \text{Aut}(G)\). In this paper, the authors consider the group \(G=\mathbb Z /a \rtimes (\mathbb Z \times Tl_2(\mathbb F_p))\) of the family VI. In particular, for this group \(G\) they determine the group \(\Aut(G)\) and compute the number of distinct homotopy types of spherical space forms of \(X(2dn-1)\), where \(2d\) is a period. Moreover, they study the group \({\mathcal E}(X)\) of homotopy self-equivalences of \(X\) for the orbit space \(X=X(2dn-1)/G\).
scientific article; zbMATH DE number 5625472
scientific article