18
$\begingroup$

Suppose that $M$ is a closed Riemannian manifold: one can construct the so called Laplace-Beltrami operator on $M$. Its spectrum contains some information of the underlying manifold: for example its dimension is encoded by the asymptotic behaviour of eigenvalues. As far as I know, there was a problem whether two isospectral closed manifolds are isometric-the negative answer was found by Milnor. So from this story I suspect that people believed that the spectrum of Laplacian will contain a lot of (geometric) information about the underlying manifold. My question is the following:

Is it possible to extract the information about the fundamental group of $M$ from the spectrum of Laplace operator?

$\endgroup$

3 Answers 3

19
$\begingroup$

It is known that the isomorphism class of $\pi_1(M)$ is not determined by the spectrum of the Laplace operator. Indeed, in 1980 Marie-France Vigneras constructed isospectral nonisometric hyperbolic $3$-manifolds. These manifolds must have non-isomorphic fundamental groups by Mostow's rigidity theorem (which implies that an isomorphism of fundamental groups would have to be induced by an isometry).

$\endgroup$
6
$\begingroup$

To add a more general, optimistic answer to the question in the title, rather than the question in the question --- while the isomorphism class of $\pi_1(M)$ is not determined by the Laplace spectrum, it is indeed possible to extract topological information.

Let $M$ be a $d$-dimensional Riemannian manifold with Laplace spectrum $\lambda_k$. McKean and Singer showed that the heat trace $Z(t) = \sum_{k=0}^\infty e^{-\lambda_k t}$ has the short-time asymptotic expansion $$ (4\pi t)^{d/2}Z(t) = \operatorname{Vol}(M) + \frac{t}{3} \int_M (\mbox{scalar curvature}) + \frac{t^2}{180}\int_M (10A - B + 2C) + o(t^3) $$ where $A$, $B$, and $C$ are polynomials in the curvature tensor. They observe that in the case of $d=2$, by Gauss-Bonnet, the second coefficient is a multiple of the Euler characteristic. I am not sure what one can say about higher-dimensional manifolds using Chern-Gauss-Bonnet.

As a second example, Cheeger and Muller independently proved that the analytic torsion of $M$ is equal to its Reidemeister torsion. The analytic torsion is the zeta-regularized determinant of the Laplacian acting on differential forms, while the Reidemeister torsion is defined in terms of a unimodular representation of $\pi_1$ and twisted homology. (Sorry I'm not more precise, I haven't thought about this in some time. Liviu Nicolaescu has good notes on this subject, which I recall were helpful to me.)

For further reading, check out Chavel's lovely book Eigenvalues in Riemannian Geometry.

$\endgroup$
3
$\begingroup$

When $M$ is negatively curved, and especially when the curvature is constant, the distribution of the eigenvalues tells something about the distribution of lengths of closed geodesics. This is because given a closed geodesic, you can construct an approximate eigenfunction. The seminal work was by A. Selberg (trace formula), improved by P. Sarnak in his PhD thesis. There have been a lot of contributors for the general theory, including Y. Colin de Verdière and more recently N. Anantharaman.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.