Section: New Results

Can you hear the homology of 3-dimensional drums?

Participant : Aurel Page.

In [16], A. Bartel and A. Page describe all possible actions of groups of automorphisms on the homology of 3-manifolds, and prove that for every prime p, there are 3-dimensional drums that sound the same but have different p-torsion in their homology. This completes previous work  [42] by proving that the behaviour observed by computer experimentation was indeed a general phenomenon.

More precisely: if M is a manifold with an action of a group G, then the homology group H1(M,) is naturally a [G]-module, where [G] denotes the rational group ring. Bartel and Page prove that for every finite group G, and for every [G]-module V, there exists a closed hyperbolic 3-manifold M with a free G-action such that the [G]-module H1(M,) is isomorphic to V. They 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 pP, the p-power torsion subgroups of H1(N,) and of H1(N',) have different orders. They 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-Müller formula, but they 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.