Talk:Mapping class group

message: 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)
replies: text: 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 Homeo_0(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)

text: 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)
thread_title:  Vagueness comments 

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

message: Let f be isotopic to the identity. Why is its inverse also isotopic to the identity? HannoBecker (talk) 15:25, 25 December 2008 (UTC)
replies: text: This is a nice exercise.  Think -- what is the definition of Homeo_0?  Best, Sam nead (talk) 13:12, 28 December 2008 (UTC)
thread_title:  Homeo0 closed under inversion? 

message: FYI, the example of the mapping class group of the torus (S^1)^n is incorrect.  This was recently pointed out on Math OverFlow.  IMO the current definition of mapping class group is perhaps too rigid and does not accurately reflect it's usage in the literature.  I think that is the heart of the problem. Rybu (talk) 00:18, 18 August 2010 (UTC)
thread_title:  Page is wrong 

message: In the last section Mumford's conjecture is mentioned although the section says that the rational cohomology ring of the stable mapping class group was conjectured. This is an empty statement, unfortunately I do not know the correct statement of this conjecture so cannot correct it.   — Preceding unsigned comment added by 129.67.185.183 (talk) 14:27, 11 February 2015 (UTC)
thread_title:  Mumford's conjecture 

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