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 Jan 14 awarded Popular Question Apr 25 awarded Disciplined Nov 10 awarded Yearling Sep 24 awarded Autobiographer Jan 7 awarded Nice Question Dec 24 awarded Critic Oct 12 awarded Supporter Oct 12 accepted Why is the mapping class group of hyperbolic manifolds finite? May 30 asked Why is the mapping class group of hyperbolic manifolds finite? Apr 25 comment Heegaard splitting, equivalent homeomorphisms, mapping class group of genus n-torus wow, this is what I was looking for! Thank you sooooo much!! Apr 23 comment Heegaard splitting, equivalent homeomorphisms, mapping class group of genus n-torus [continued] What can we say about the two resulting manifolds? Even if we take f and g in the same double coset, they do not in general extend to automorphisms of the manifolds, so we don't get an isomorphism between them. Moreover, if they give the same manifold, do they give equivalent splittings? This is the kind of questions I'm thinking about. Apr 23 comment Heegaard splitting, equivalent homeomorphisms, mapping class group of genus n-torus I didn't find anything at that page...anyway I already knew that book, so I think you are referring to the article of Birman (maybe we have different page numberings). The point is I'm not satisfied with this :) I mean: by Birman, we know that if a manifold admits two equivalent splittings, than they are obtain from gluing maps which are in the same double coset. But what about the converse? Suppose we are given two genus g handlebodies and we want to glue them along their boundaries to obtain a 3-manifold, and we use two different maps, say $f$ and $g$. Apr 20 comment Heegaard splitting, equivalent homeomorphisms, mapping class group of genus n-torus Hi Maxime, can you please give me a reference for the first part of your answer ("You obtain the same Heegaard splitting iff your homeomorphisms are in the same double coset H\M/H where H is the subgroup of mapping classes that extend to the handlebody.") Moreover, I can't catch which is the relation between saying that two splitting give the same manifold (up to homeomorphism) and saying that two splitting are equivalent splittings for the same manifold. Someone can help? thanks in advance Mar 24 awarded Scholar Mar 24 comment Why is Casson's invariant worth studying? ok thank you, and sorry..it's my first question on MathOverflow Mar 24 accepted Why is Casson's invariant worth studying? Mar 22 comment Why is Casson's invariant worth studying? Wow thank you so much..you give me lots of things to think about! thank you again for your answer! Mar 17 awarded Student Mar 17 asked Why is Casson's invariant worth studying?