**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 $G=\mathrm{Diff}(M)$ 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, $\mathrm{Homeo}(M\_i)$ (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


*Banyaga, Augustin*, The structure of classical diffeomorphism groups, Mathematics and its Applications (Dordrecht). 400. Dordrecht: Kluwer Academic Publishers. xi, 197 p. (1997). [ZBL0874.58005](https://zbmath.org/?q=an:0874.58005).



To which extent can one recover a manifold from its group of homeomorphisms
Answer is: Yes, one can recover $M$ if it is a compact manifold. See [J. V. Whittaker: On Isomorphic groups and homeomorphic spaces, Annals of Math 1963.](http://dl.dropbox.com/u/5188175/whit.pdf)


**EDIT** Actually, one knows a lot more, see, for example 
Tomasz Rybicki 
Journal: Proc. Amer. Math. Soc. 123 (1995), 303-310. 
MSC: Primary 58D05; Secondary 17B66, 22E65, 57R50 
MathSciNet review: 1233982 


And references therein...


**ANOTHER EDIT**


A quite different proof of a stronger theorem (actually a large set of theorems) than Whittaker's (actually, Whittaker's paper seems to be rather badly written) is given by Matatyahu Rubin in 
Rubin, Matatyahu(3-SFR)
On the reconstruction of topological spaces from their groups of homeomorphisms. 
Trans. Amer. Math. Soc. 312 (1989), no. 2, 487–538. 


