Group representations in the homology of 3-manifolds

TL;DR

This paper demonstrates that for any finite group, one can realize any rational G-module as the homology of a hyperbolic 3-manifold with a free G-action, and explores spectral geometry implications.

Contribution

It constructs hyperbolic 3-manifolds with prescribed G-module homology and reveals that homology does not determine fixed point data, contrasting with 2D cases.

Findings

Existence of hyperbolic 3-manifolds with prescribed G-module homology.

Construction of strongly isospectral manifolds with different torsion homology.

Homology of 3-manifolds with G-action does not encode fixed point information.

Abstract

If M is a manifold with an action of a group G, then the homology group H_1(M,Q) is naturally a Q[G]-module, where Q[G] denotes the rational group ring. We prove that for every finite group G, and for every Q[G]-module V, there exists a closed hyperbolic 3-manifold M with a free G-action such that the Q[G]-module H_1(M,Q) is isomorphic to V. We give an application to spectral geometry: for every finite set P of prime numbers, there exist hyperbolic 3-manifolds N and N' that are strongly isospectral such that for all p in P, the p-power torsion subgroups of H_1(N,Z) and of H_1(N',Z) have different orders. We also show that, in a certain precise sense, the rational homology of oriented Riemannian 3-manifolds with a G-action "knows" nothing about the fixed point structure under G, in contrast to the 2-dimensional case. The main geometric techniques are Dehn surgery and, for the spectral application, the Cheeger-M\"uller formula, but we also make use of tools from different branches of algebra, most notably of regulator constants, a representation theoretic tool that was originally developed in the context of elliptic curves.