Remarks on the cohomology of finite fundamental groups of 3-manifolds

TL;DR

This paper computes the cohomology rings of finite groups acting freely on S^3 and their associated 3-manifolds, using explicit resolutions and chain approximations, with applications discussed.

Contribution

It provides explicit 4-periodic resolutions and chain homotopies for specific finite groups acting on S^3, advancing understanding of their cohomology.

Findings

Explicit cohomology computations for groups G acting on S^3

Construction of chain approximations to the diagonal

Applications to the topology of spherical space forms

Abstract

Computations based on explicit 4-periodic resolutions are given for the cohomology of the finite groups G known to act freely on S^3, as well as the cohomology rings of the associated 3-manifolds (spherical space forms) M = S^3/G. Chain approximations to the diagonal are constructed, and explicit contracting homotopies also constructed for the cases G is a generalized quaternion group, the binary tetrahedral group, or the binary octahedral group. Some applications are briefly discussed.