I’m not sure exactly the best way to pose an open problem about open book decompositions and Heegaard splittings, but here’s the idea: Recall that an open book decomposition for a 3-manifold is a link in the 3-manifold and a surface bundle structure on the link complement such that each leaf of the bundle is a spanning surface for the link. The closure of the union of any two leafs form a Heegaard surface. The problem I want to suggest is, given a Heegaard splitting, to understand the set of open book decompositions that induce a Heegaard splitting in the same isotopy class. (Note that a Heegaard splitting induced by an open book has Hempel distance at most two, so it’s easy to find splittings that aren’t induced by any open books.)

The most immediate reason to want to understand this set of open book decompositions is that spinning around the circle factor of the surface bundle determines an automorphism of the Heegaard splitting (and as you may have noticed I’m currently rather interested in automorphisms of Heegaard splittings.) So understanding the collection of open book decompositions would tell us something about Heegaard splitting’s the mapping class group.

But there’s another reason that the connection between open books and Heegaard splittings is interesting. There is a very strong, and reasonably well understood connection between open books and contact structures, due to the fact that the plane field defined by an open book decomposition can be deformed into a canonical contact structure. So open book decompositions may be a good way to find some relation ships between contact structures and Heegaard splittings. (For example one might ask if the existence of a tight contact structure implies the existence of a strongly irreducible Heegaard splitting or vice versa.)

The way that open books interact with contact structures and Heegaard splittings is slightly different. Given an open book decomposition, there is a construction called Hopf plumbing that essentially consists of connect summing the open book decomposition with a copy of a Hopf link in the 3-sphere. This decreases the Euler characteristic of the leaf by one and changes the number of components of the link by one. It affects the induced Heegaard splitting by adding a stabilization. The effect on the contact structure depends on whether the hopf link has positive or negative linking number. I think the way it works (though someone correct me if I’m getting this wrong) is that a positive Hopf plumbing has no effect on the contact structure, while a negative plumbing will turn a tight contact structure into an over-twisted one.

Giroux and Goodman [1] have used the correspondence between open books and contact structures, and the fact that overtwisted contact structures are determined entirely by their homology class, to show that open book decompositions that determine the same homology class in a given 3-manifold are stably equivalent under Hopf plumbing. They note that this is related to stable equivalence for Heegaard splittings. Trying to prove stable equivalence of open books via their connection to Heegaard splittings turns out to be more difficult because it’s unclear how to calculate the homology class of the open book from its spine in the Heegaard surface. (I vaguely remember seeing a preprint related to this, but I wasn’t able to find it when I went back and looked.) But the point is, there seems to be something there to explore.

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