Torsion homology and regulators of Vignèras isospectral manifolds

Virtual talk.

The problem of isospectral manifolds was popularised by Mark Kac in the 60's under the formula "Can you hear the shape of a drum?''.  We say that two compact Riemannian manifolds M and N are \(k\)-isospectral if the Laplace operators acting on \(k\)-forms of \(M\) and \(N\) have the same spectrum counted with multiplicity. Two manifolds that are \(k\)-isospectral for all \(k\) must share many invariants, such as dimension, volume and Betti numbers, but they are not necessarily isometric: the first counter-examples in dimension 3 were constructed by Vign\'eras using arithmetic manifolds. The Cheeger-Mueller formula implies that a ratio \(R\) of regulators of \(M\) and \(N\) equals the ratio of size of torsion in the first homology group, so it is natural to ask whether we must have \(R=1\), and if not which primes may appear in the factorisation of \(R\). In the case of Vignèras manifolds, we give a more arithmetic interpretation for these numbers, and give sufficient criteria for the valuation of \(R\) at a given prime \(p\) to be \(0\) and for the \(p\)-primary torsion subgroup of \(H_1(M)\) and \(H_1(N\)) to be isomorphic. This is joint work with Alex Bartel.

Here [3] you can find the slides to this talk.