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 , there are 3-dimensional drums that sound the same
but have different -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 is a manifold with an action of a group , then the
homology group is naturally a -module, where
denotes the rational group ring. Bartel and Page prove that for every finite
group , and for every -module , there exists a closed hyperbolic
3-manifold with a free -action such that the -module
is isomorphic to . They give an application to spectral geometry: for every
finite set of prime numbers, there exist hyperbolic 3-manifolds
and that are strongly isospectral such that for all , the
-power torsion subgroups of and of have different
orders. They also show that, in a certain precise sense, the rational
homology of oriented Riemannian 3-manifolds with a -action "knows" nothing
about the fixed point structure under , 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.