Fundamental Domain and Cellular Decomposition of Tetrahedral Spherical Space Forms

Abstract

Given a free isometric action of the binary tetrahedral group on a (4n − 1)-dimensional sphere, we obtain an explicit finite cellular decomposition of the sphere, equivariant with respect to the group action. A cell decomposition of the correspondent spherical space form and an explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group of the space form follows. In particular, the construction provides a simple explicit 4-periodic free resolution for the binary tetrahedral group.