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I got rid of the link to topological rigidity since that entry is different than is what is meant by Waldhausen's proof of topological rigidity for Haken manifolds.
Topological rigidity is actually an ambiguous term, meaning different things in different contexts (such as in this article and the link above).
In the Waldhausen case, what is meant is that homotopy equivalent implies homeomorphic and homotopic homeomorphisms are isotopic. -- C S 02:13, Sep 8, 2004 (UTC)
My mistake. I have added (at the end) a mention of Johannson's work (which I had muddled up in my mind with Waldhausen's). Probably need a whole set of articles devoted to the various kinds of rigidity in mathematics. I will also rewrite the mapping class groups article a bit.-- sam Wed Sep 8 10:09:23 EDT 2004
I have a question regarding:
What is meant by the last sentence? In particular, what does "the hyperbolic case" mean? In terms of hyperbolic Haken manifolds, they are atoroidal Haken manifolds. So this would appear already covered by Johannson. "Subsumed", to me, implies that there is something more general. But I'm not sure how Thurston's geometrization theorem coupled with Mostow rigidity is more general.
Perhaps you are referring to Thurston's geometrization conjecture instead? So something like, "This work is subsumed by Thurston's geometrization conjecture (if true). If a 3-manifold is atoroidal with infinite fundamental group, then it is hyperbolic and thus has finite mapping class group." -- C S 15:24, Sep 8, 2004 (UTC)
Gack -- perhaps instead of "subsumed" it would be better to say "recovered", or "reproved". I am pretty sure that Johannson only needs geometrically atoriodal - so his work covers atoroidal Haken Seifert fiber spaces. Hmmm. But perhaps you can dig around and find a simpler proof (something that Seifert would have thought up) for those manifolds. Then "subsumed" would be fair - the hyperbolic case is the hard one, after all!
Not sure I understand your last comment - remember that Johannson assumed Hakenness.-- sam Thu Sep 9 03:00:01 EDT 2004
Perhaps more interesting would be for us to rewrite this page - it is somewhat vague, I think.-- sam Thu Sep 9 03:03:34 EDT 2004