# 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 $M$?

• 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
• @Igor: It's true that not all homeomorphisms are smooth, but homeomorphism and diffeomorphism groups are always comparable as there's an injection from one to the other. @Hugo: in general MO isn't a place for "tell me about X" type questions. Question-askers have an obligation to be fairly specific. Other than your first "what can we say in general about..." question, it appears your more specific questions have been answered. – Ryan Budney Dec 22 '10 at 16:52

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