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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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    $\begingroup$ I find the question much too broad. $\endgroup$ – Daniel Moskovich Dec 22 '10 at 14:26
  • $\begingroup$ Indeed Daniel, the question is quite broad! $\endgroup$ – Hugo Chapdelaine Dec 22 '10 at 14:32
  • $\begingroup$ (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 $\endgroup$ – Ryan Budney Dec 22 '10 at 15:45
  • $\begingroup$ 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. $\endgroup$ – Igor Rivin Dec 22 '10 at 16:25
  • $\begingroup$ @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. $\endgroup$ – Ryan Budney Dec 22 '10 at 16:52
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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.

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  • $\begingroup$ Don't you need some hypotheses to conclude simplicity? Homeo(surface) isn't simple. $\endgroup$ – Autumn Kent Dec 22 '10 at 16:44
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    $\begingroup$ Presumably Rivin is talking about the identity component of the group, not the whole homeomorphism group. $\endgroup$ – Ryan Budney Dec 22 '10 at 16:46
  • $\begingroup$ Yes, that's correct, typing too fast :( $\endgroup$ – Igor Rivin Dec 22 '10 at 16:51
  • $\begingroup$ The appendix of Freedman and Calegari paper (Distortion in transformation groups) is from "Yves de Cornulier". Thanks a lot @Igor for the reference. $\endgroup$ – Hugo Chapdelaine Dec 22 '10 at 17:08
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$\pi_0(Aut(M))$ is the mapping class group of $M$. The Wikipedia article has some good examples.

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