Talk:Heegaard splitting

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In the applications and connections section it might also be nice to have a summary of the thread started by Mark Lackenby towards understanding the virtual Haken conjecture via coverings, weak irreducibility and group theory. In a similar vein it might also be nice to discuss Heegaard splittings via the complex of curves.

An important early theorem is the Reidemeister Singer theorem, this could lead into a discussion of Heegaard diagrams, which could make the whole affair a lot more concrete because you could draw some pictures of three-manifolds.

Another fundamental result (and maybe one of the fundamental results about three manifolds) is Haken's lemma. Its completely missing from the discussion.

The biggest modern idea that goes without mention is Rubinstein's concept of a sweep, which had implications both in the study of minimal surfaces in three manifolds, and through the work of Rubinstein and Scharlemann on stabilizing Heegaard splittings on the theory of Heegaard splittings. --User:Topoman

Yes, I've been wanting to add Lackenby's Heegaard gradient stuff for a while. The curve complex stuff is getting a lot of attention so that's a good idea. But before we do that, we should lay the groundwork by doing the classical results, as you say, like Haken's lemma. This would lead to Casson-Gordon stuff, untelescoping splittings, and to the notion of thin position by Scharlemann-Thompson.

Interesting, I didn't know it was Rubinstein who first used the concept of a sweepout in this context (I assume that's what you mean by "sweep"). But yes, that definitely ought to be mentioned. Maybe some stuff also about his work (with Pitts) on using sweepouts (via minimax methods) to find minimal surfaces, such as those which are also strongly irreducible Heegaard surfaces.

Hmmm...this sounds like a lot of work! With several people working together though, it should be doable within say, a year(?). --Chan-Ho Suh 22:00, Jan 3, 2005 (UTC)

Rubinstein (and Pitts?) were the first to use sweepouts in an interesting way, I think. Previous work used "sweepouts" where the generic levels all have the same genus and where the generic levels (almost) foliate the manifold. The classic approach is to take this very nice sweepout and apply thin position, eg the classification of splittings of surface cross [0,1]. Rubinstein and Pitts allow surface doing the sweeping to pinch and "back up". Also, if you take the standard sweepout of S^3 by spheres and map it to RP^3 then they would still call that a sweepout, I think. Doing thin position with respect to such a sweepout would be harder, I am sure...--Sam nead 23:13, 14 August 2005 (UTC)

[edit] Heegaard Floer Homology

I wikilinked the section Heegaard Floer homology (Header and first occurence in the text) to the section of the same name in the article Floer homology - not being quite sure if this is 100% Wikipedia style. Rolf of: Erkabo 09:21, 14 November 2005 (UTC)

That's fine. The idea is that the reader should be able to tell the main article is elsewhere. One thing that is done sometimes is to put the comment "See also the main article for Heegaard Floer homology" right under the header. But that's probably not necessary in this case since there really is no main article, but a section elsewhere for now. --Chan-Ho(talk) 10:37, 14 November 2005 (UTC)

Chan-Ho, should I do the same - vice versa - in the article Floer homology? Say yes and I'll do it. Rolf of Erkabo 11:11, 14 November 2005 (UTC)

No, it's better you don't. That way people will expand the Floer homology article. The info in this Heegaard splitting article should mainly be on Heegaard splitting with some mention of applications. It should be considered a brief summary with the link you made leading to the main article. By the way, I'm still digesting the information you sent me. But I hope to expand and make the corrections to Andreas' article in the next couple weeks. --Chan-Ho (Talk) 12:18, 14 November 2005 (UTC)