Let $M$ be a $n$ dimensional manifold. Let $Aut(M)$ be the group of homeomorphisms
of $M$ viewed as a topological group under the compact-open topology. 


What can we say in general about $Aut(M)$? 


1.For example, what is $\pi\_0(Aut(M))?$


2.Is $Aut(M)$ homotopy equivalent to a space which can be describe
in terms of a CW complex structure of $M$? 



On the group of homeomorphisms of a manifold
A lot has been said about homeomorphism and diffeomorphism groups of spaces (more diffeo), starting with von Neumann and Ulam in the mid 40s (probably before then, as well), and until today. I would advise looking at papers by D.B.A. Epstein in the late sixties, John Mather in the early seventies, and a cool paper by D. Calegari and M. Freedman a couple of years ago (with an even cooler appendix by Y. de Cournouiller). The key fact (discovered very early on) is that these groups are perfect (equal to their commutator subgroup) and simple. It is an interesting question (which seems to be due to myself, but it is hard to believe that von Neumann/Ulam did not already wonder about this) whether every homeomorphism/diffeomorphism is, in fact, a commutator.


