Question. Suppose that $M$ is a closed connected topological manifold and $G$ is its group of homeomorphisms (with compact-open topology). Does $G$ (as a topological group) uniquely determine $M$?
One can ask the same question where we regard $G$ as an abstract group (ignoring topology), replace topological category by smooth category (here one can equip $G=Diff(M)$ with a finer structure of a Frechet manifold), varying degree of smoothness, dropping compactness assumption, recovering $M$ up to homotopy, etc.
I do not know how to answer any of these questions. I do not even know if one can recover the dimension of $M$ from its group of homeomorphisms. In low dimensions, or assuming that $M$ has a locally-symmetric Riemannian metric, and if $dim(M)$ is given, I know few things. For instance, among 2-dimensional manifolds one can recover $M$ from $G$ since $G/G_0$ is the mapping class group $Mod(M)$ of $M$ and one can tell the genus of $M$ from maximal rank of free abelian subgroups of $Mod(M)$. Same for, say, closed hyperbolic manifolds with non-isomorphic isometry groups. However, given, for instance, two closed hyperbolic 3-manifolds $M_1, M_2$ with trivial isometry groups, I do not know how to distinguish $M_i$'s by, say, $Homeo(M_i)$ (the problem reduces to a question about homeomorphism groups of the unit ball commuting with $\pi_1(M_i)$, $i=1,2$, but I do not see how to solve it).
Update: Results quoted by Igor and Martin give the complete answer in topological and smooth category in the strongest possible form (much more than I expected!). Positive answer is also known in the symplectic category, but, apparently, is open for contact manifolds and their groups of contactomorphisms.
Another reference in the smooth case, sent to me by Beson Farb is the book by Augustin Banyaga, "The structure of classical diffeomorphism groups."