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

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) MD 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. 57M05, 57M60; 20J06 The structure of the cohomology rings of 3‐manifolds is an area to which Heiner Zieschang devoted much work and energy, especially from 1993 onwards. This could be considered as part of a larger area of his interest, the degrees of maps between oriented 3‐manifolds, especially the existence of degree one maps, which in turn have applications in unexpected areas such as relativity theory (cf Shastri, Williams and Zvengrowski [41] and Shastri and Zvengrowski [42]). References [1; 6; 7; 18; 19; 20; 21; 22; 23] in this paper, all involving work of Zieschang, his students Aaslepp, Drawe, Sczesny, and various colleagues, attest to his enthusiasm for these topics and the remarkable energy he expended studying them.