Talk:Haken manifold

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:

Also worthy of note is Klaus Johannson's proof that atoroidal Haken three-manifolds have finite mapping class groups. In the hyperbolic case, this work is subsumed by the combination of Mostow rigidity with Thurston's theorem.

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

I'm not sure what the confusion is. I'm saying that the geometrization conjecture does "subsume" earlier results, like Johannson's. --C S 15:57, Oct 15, 2004 (UTC)

Ok -- that is not correct -- Johannson proves finiteness of the mapping class group for Haken, atoroidal, three-manifolds. Now, most of these are hyperbolic and so geometrization and Mostow rigidity gives another proof. But some of them are not hyperbolic but instead Seifert fibred spaces. We know these are geometric -- how do we use that to prove finite mapping class group? All the best,--sam Sun Oct 17 20:15:39 EDT 2004

Sorry for taking so long to respond. Anyway, Johannson does not do what you say. He proves the result for Haken, atoroidal, anannular 3-manifolds. Also, I don't think he allows discs to be essential surfaces. So the case of Seifert fiber spaces does not occur.--C S 00:58, Oct 30, 2004 (UTC)

Um, how do disks come into it? SFS's have incompressible boundary... (Hmmm. Except for the solid torus?) Anyway, I have checked out Johannson's first book from the library, and will try to do some reading. Best,--sam Wed Nov 3 10:32:17 EST 2004

Well, clearly the solid torus is Haken with no essential tori or annuli and we don't want it! That's why I made my disc comment. And yes, for an (orientable) SFS to have compressible boundary, it must be a fibered solid torus. In any case, my point was that SFSs are not covered by Johannson's result (as in his paper I found on MathSciNet). --C S 00:43, Nov 4, 2004 (UTC)

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

Yes, a rainy day project....it hasn't been raining too often where I live though :-) --C S 15:57, Oct 15, 2004 (UTC)