2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/226769Computations 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.This is the version published by Geometry & Topology Monographs on 29 April 2008Algebraic TopologyGroup TheoryGeometric Topology57M05, 57M60, 20J06text