Let $(M,g)$ be a (say closed) Riemannian manifold. One can try to understand the geometry/topology of $(M,g)$ by studying the eigenvalues of the Laplacian (this I guess has two versions: when considering the Laplacian on functions only, or on differential forms. feel free to answer about whichever).

Sometimes (for example for nice bounded planar domains) the topology of the manifold is completely determined by the spectrum of the Laplacian (on functions).

Also, by the Hodge theorem, for an arbitrary closed Riemannian manifold all of the Betti numbers can be read off the spectrum of the Laplacian (on forms). Sadly, it is well known that there are examples of isospectral but not isometric Riemannian manifolds.

Q1: Are there examples of pairs of Riemannian manifolds which are isospectral but not diffeomorphic/homeomorphic/homotopy equivalent? If so, what is the simplest one?

assuming the answer to Q1 is "Yes" or "It is not known":

Q2: What are the best positive results in this direction? How much can I know about the topology of $M$ only from my knowledge of the spectrum of the Laplacian?

I'm sorry if the question is too vague, but still any information will be nice.

Progress in Inverse Spectral Geometry(ed. S. Anderson and M. Lapidus), published by Springer in its "Trends In Mathematics" series. $\endgroup$ – Vesselin Dimitrov May 17 '18 at 13:39