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 $H_1(M,\mathbb{Q})$ is naturally a $\mathbb{Q}\left[G\right]$-module, where $\mathbb{Q}\left[G\right]$ denotes the rational group ring. Bartel and Page prove that for every finite group $G$, and for every $\mathbb{Q}\left[G\right]$-module $V$, there exists a closed hyperbolic 3-manifold $M$ with a free $G$-action such that the $\mathbb{Q}\left[G\right]$-module $H_1(M,\mathbb{Q})$ 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 $N^{\text{'}}$ that are strongly isospectral such that for all $p\in P$, the $p$-power torsion subgroups of $H_1(N,\mathbb{Z})$ and of $H_1(N^{\text{'}},\mathbb{Z})$ 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.