Talk:Mapping class group

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Comment on removing 'vagueness'. It's never really easy to defend any particular looser form of words relating to (pure) mathematics. But on the oher hand, if all the hand waving gets squeezed out, the overall impression is of rapid-fire density. So, I'm always going to stick up for rough explanations in introductory stuff.

Charles Matthews 10:24, 7 Sep 2004 (UTC)

Fair enough -- I was just worried about the term "internal symmetry group" which isn't defined anywhere. I hope the new version is ok. sam Tue Sep 7 09:31:35 EDT 2004

Under what circumstances can Homeo(X) be made into a topological group such that Homeo0(X) is the identity component? Is this true whenever X is locally compact Hausdorff with the compact-open topology on the homeomorphism group? Just curious. -- Fropuff 16:01, 6 October 2005 (UTC)

Ack -- what a question! Without thinking deeply, I believe that the definitions in the article are mostly aimed at manifolds. For a general topological space the group Homeo(X) may not be large enough to be interesting (think about the example of a graph). Instead, one should think about homotopy equivalences mod homotopy. (For closed orientable surfaces this new definition is the same as the old one.) As for your question: "all" that needs to be checked is that a path in Homeo(X) gives an isotopy and reversely. Forward direction: suppose that \gamma \from [0,1] \to Homeo(X) is a path. Define \Gamma(x, t) = \gamma(t)(x). So we have to show that \Gamma \from X \cross [0,1] \to X is continuous. Take a point (x, t) in X \cross [0,1] and let V be any neighborhood of \Gamma(x,t). We have to find a neighborhood U of x and an interval (t-a, t+a) so that \Gamma(U \cross (t-a, t+a)) lands in V. Gotta go, more later. -- Sam nead 14:32, 14 July 2006 (UTC)
  • It was first observed by Birman and Chillingworth that the group presentation of MCG(N_3) is < a,b,j: aba=bab, (aba)^4=1, j^2=1, jaj=a^{-1}, jbj=b^{-1}> in 1972, without saying that this group is GL_2(Z), see page 448 on {On the homeotopy group of a non-orientable surface}, Proc. Camb. Phil. Soc. 71 (1972), 437-448. Juan Marquez 05:59, 16 October (UTC)

[edit] great work

Really superb page, nice work! MotherFunctor 07:36, 19 May 2006 (UTC)

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