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?