Space Forms and Group Resolutions: the tetrahedral family

TL;DR

This paper constructs cellular models for spherical space forms using orbit polytopes, applies them to the tetrahedral group family, and computes their homology, cohomology, and Reidemeister torsions.

Contribution

It introduces a method to derive resolutions and invariants of spherical space forms from orbit polytopes for finite groups, specifically applied to the tetrahedral family.

Findings

Homology groups of the space forms are explicitly computed.

Cohomology rings of the space forms are determined.

Reidemeister torsions are calculated for the tetrahedral family.

Abstract

The orbit polytope for a finite group G acting linearly and freely on a sphere S is used to construct a cellularized fundamental domain for the action. A resolution of the integers over G results from the associated G-equivariant cellularization of S. This technique is applied to the generalized binary tetrahedral group family; the homology groups, the cohomology rings and the Reidemeister torsions of the related spherical space forms are determined.