Talk:Mapping class group

Vagueness comments

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 $Home{o}_0(X)$ 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)

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)

great work

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

Homeo0 closed under inversion?

Let f be isotopic to the identity. Why is its inverse also isotopic to the identity? HannoBecker (talk) 15:25, 25 December 2008 (UTC)

This is a nice exercise. Think -- what is the definition of Homeo_0? Best, Sam nead (talk) 13:12, 28 December 2008 (UTC)

Page is wrong

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)

I patched up the page a little so that the big errors and omissions of notation aren't quite as glaring. What do people think? Rybu (talk) 01:12, 18 August 2010 (UTC)

Mumford's conjecture

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)

Merger proposal

Template:Discussion top There is another page about basically the same subject (just restricted to surfaces) at Teichmüller modular group. The two are not even linked to each other. --129.132.146.194 (talk) 11:23, 27 July 2011 (UTC)

I think that they should be merged. In fact, Teichmueller modular group is not heard among experts as mapping class group. I would suggest having one page `Mapping class group' and have a redirect from Teichmueller modular group to Mapping class group. As you have noted, the use of Teichmueller doesn't really apply to higher dimensional manifolds. Hence it should be made clear that this term only applies in the 2-dimensional case. The action of mapping class group on Teichmueller space should have a place on this unified page. (Note that there is no evidence that Teichmueller, the mathematician, ever considered the mapping class group.) (Lost-n-translation (talk) 01:56, 29 July 2011 (UTC))

The concept in both articles is the same. But the (rather simple) concept of a mapping class groups has no intrinsic connection with Teichmüller space, which is a complicated concept involving conformal equivalence classes etc.

Conclusion: The article "Teichmüller modular group" should redirect to "Mapping class group", where "Teichmüller modular group" should be mentioned 'solely as being a synonym for "Mapping class group". Daqu (talk) 09:48, 1 December 2013 (UTC)

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