Bill Thurston is dead at age 65.

Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years.   I don’t think it’s possible to overstate the revolutionary impact that he had on the study of geometry and topology.  Almost everything we blog about here has the imprint of his amazing mathematics.    Bill was always very generous with his ideas, and his presence in the community will be horribly missed.    Perhaps I will have something more coherent to say later, but for now here are some links to remember him by:

• Wikipedia.
• 2010 lecture on The mystery of 3-manifolds.
• On proof and progress in mathematics.

Generalizations of open books

I’m going to take a break from data topology for this post and write about an interesting construction that I heard Jeremy Van Horn-Morris talk about at the Georgia Topology conference at the beginning of the summer. I should admit that it took me a while to appreciate this definition of a generalized open book decomposition because they only occur in toroidal 3-manifolds with very specific JSJ decompositions. However, they come out of a very natural generalization of 4-dimensional Lefschetz fibrations in which the 3-manifold arises as the boundary of the 4-manifold. These were first developed by Jeremy, Sam Lisi, and Chris Wendl, in a preprint that is still being written. Jeremy and Inanc Baykur [1] also use this construction to produce contact structures that disprove a number of former conjectures, so even though these 3-manifold are not hyperbolic, they are interesting from the perspective of contact topology.

Contact structures vs. Heegaard splittings

I just got back from the Georgia topology conference. While I was there, I talked to a number of contact topologists, and the conversations mostly revolved around open book decompositions. I’ve mentioned the links between Heegaard splittings, open books and contact structures previously, but I wanted to reiterate my feeling that his should be an interesting area to study. It’s very easy to go from an open book to a Heegaard splitting or from an open book to a contact structure. Conversely, every contact structure induces a whole family of open books, but there is no good way (so far) to get from a Heegaard splitting to an open book. Many Heegaard splittings (in particular, high distance ones) don’t come from any open book, while others could come from lots of unrelated open books.

I don’t have anything else to say about possible connections, other than to suggest studying the way a sweep-out for a Heegaard splitting passes through the contact structure. One could, for example, consider the family of foliations of the surfaces that come from intersecting the sweep-out surfaces with the contact planes. There are probably better ways to analyze the intersection of the sweep-out surfaces and the contact planes as well, but I don’t know what they are. I’m going to add the following question to the open problems page, which I think roughly captures the problem: Is there a connection between the existence of a tight/fillable/etc. contact structure and the existence of a high distance Heegaard splitting? In other words, if a 3-manifold admits one of these types of contact structure, does that imply the existence, or perhaps rule out the existence of a high distance Heegaard splitting?