Spectral properties of the Laplace operator and topological properties

Question

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?

Answer 1

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).

Answer 2

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.

Answer 3

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.