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  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.
A Lefschetz fibration is a map from a four-dimensional manifold to the disk that is a fibration on the complement of finitely many points in the disk and has prescribed singularities at these non-regular points. The pre-image of a generic point is a surface (of the same genus at every point). At the singularities, the surfaces limit onto a pinched surface in which an essential loop has been shrunk down to a point. I think of these as a four-dimensional version of Seifert fibered spaces, though I don’t know if that’s a reasonable analogy.
The first thing to note about a Lefschetz fibration is that if you follow a loop in the base disk, you can locally patch together a surface cross an interval, then when you get back to the starting point you have to glue the initial surface to the final surface. In other words, the loop in the base disk lifts to a 3-dimensional surface bundle sitting inside the 4-dimensional manifold. In particular, if you take a small loop around one of the singularities, the monodromy of this surface bundle will be a Dehn twist along the loop in the fiber surface that is crushed to a point at the singularity. If you choose a loop that goes around a few different singularities, the monodromy will be a composition of the corresponding Dehn twists.
In particular, the boundary of the base disk is a loop that surrounds all the singular points. If the fiber surface is closed then the boundary of the 4-manifold will be the surface bundle lifted from this loop. However, if the fiber surface has boundary, then the boundary of the 4-manifold is the union of this surface bundle and something else: Each boundary loop of the fiber surface defines a circle bundle over the base disk of the Lefschetz fibration. However, there are no singularities in this bundle, so it’s just a product, i.e. a solid torus. In other words, the boundary of this four manifold is a surface bundle whose torus boundary components have been filled in with solid tori. This is a 3-manifold with an open book decomposition.
Baykur and Van Horn-Morris generalize this construction by replacing the base disk in the Lefschetz fibration with a higher genus surface, possibly with multiple boundary components. The boundary of the resulting manifold is now pieced together from two types of pieces: Each boundary loop of the base surface again defines a surface bundle with torus boundary components. Each boundary component of the fiber surface defines a circle bundle over a surface with no singularities, i.e. a product. Because these pieces have incompressible boundary tori (since the base is not a disk), they glue together to form a 3-manifold with incompressible tori. In particular, these pieces are (roughly) the components of the JSJ decomposition.
So here’s the question: What can 3-manifold topology tell us about these manifolds? I think there’s a perception that everything interesting is already known about JSJ decompositions, but this isn’t true. The JSJ machinery is very powerful, but there are interesting combinatorial questions that arise when applying it. (See, for example, my fellow blogger Ryan Budney’s paper on JSJ decompositions of knot and link complements in the three-sphere .) Bakyur and Van Horn-Morris’ work suggests there should be even more interesting questions along these lines that haven’t been asked yet.