# On the group of homeomorphisms of a manifold

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 $M$?

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I find the question much too broad. –  Daniel Moskovich Dec 22 '10 at 14:26
Indeed Daniel, the question is quite broad! –  Hugo Chapdelaine Dec 22 '10 at 14:32
(2) generally no. (1) see Landsburg's answer below. If you want to get a sense for $Aut(M)$ search MO for things like "diffeomorphism group". Also, there's this: mathoverflow.net/questions/18034/can-we-decompose-diffmxn –  Ryan Budney Dec 22 '10 at 15:45
Diffeomorphism group is quite different from the homeomorphism group, in general (and diffeomorphism group for different degrees of smoothness are quite different from one another. –  Igor Rivin Dec 22 '10 at 16:25
Thanks Ryan for the advice. –  Hugo Chapdelaine Dec 22 '10 at 16:59

$\pi_0(Aut(M))$ is the mapping class group of $M$. The Wikipedia article has some good examples.