Homotopy types of orbit spaces and their self-equivalences for the periodic groups \(\mathbb Z/a \rtimes (\mathbb Z/b \times T^\star_n)\) and \(\mathbb Z/a \rtimes (\mathbb Z/b \times O^\star_n)\) (Q2381407)

The authors study the homotopy types of the orbit spaces of free actions of finite periodic groups on homotopy spheres. Suppose that \(G\) is a finite group of period \(2d\) of the form \(\mathbb Z/a \rtimes (\mathbb Z/b \times T^\star_n)\) or \(\mathbb Z/a \rtimes (\mathbb Z/b \times O^\star_n)\). Let \(X(n)\) be an \(n\)-dimensional \(CW\)-complex of the homotopy type of an \(n\)-sphere. If \(\mu\) is a cellular action of \(G\) on \(X(n)\) then the orbit space of \(\mu\) on \(X(n)\) is denoted by \(X(n)/\mu\). The authors compute the number of distinct homotopy types \(X(2dz-1)/\mu\). They also determine the groups of self homotopy equivalences \(\mathcal{E}(X/\mu)\) of the orbit spaces \(X(n)/\mu\).