Spherical space forms--Homotopy types and self-equivalences for the groups \(\mathbb Z/a \rtimes \mathbb Z/b\) and \(\mathbb Z/a \rtimes(\mathbb Z/b \times \mathbb Q_{2^i})\)
If \(\gamma: G\times X\to X\) is a free and celluar action, where \(X\) is a CW complex homotopy equivalent to a sphere, then the quotient \(X/\gamma\) is called a spherical space form. Two spherical space forms \(X/\gamma\) and \(X'/\gamma\) are said to be equivalent if they are homeomorphic; they are homotopic (denoted \(\simeq\)) if they are homotopy equivalent. The set of equivalence classes of spherical space forms \(X/\gamma\) where \(X\simeq S^{2d-1}\) is denoted \(\mathcal{K}_G^{2d-1}\). The first main result of this paper determines the cardinality of the set \(\mathcal{K}_G^{2d-1}/\simeq\) for groups of the form described in the title, and for certain dimensions \(2d-1\); the statement is too technical to be stated here. The second main result, also very technical, describes the group of self-equivalences of spherical space forms.
scientific article; zbMATH DE number 2136483
scientific article