The basic tenet of Thurston’s approach to 3-manifolds is the idea that the topology of a 3-manifold should determine a fairly canonical geometric structure on the manifold (modulo some technicalities). This suggests that there should be a dictionary for translating topological features of the manifold into geometric features, an idea that has been called WYSIWYG (what you see is what you get) topology. (I think Steve Kerckhoff coined the term.) While the Theorems relating the topology of Heegaard splittings to geometry are still a bit coarse, they give a very nice intuitive picture that I think is useful, and I would like to describe this picture below.

I want to start with the ideas of minimal surfaces and harmonic surfaces, which are the basis for Lackenby’s work on Heegaard gradient and the virtually Haken conjecture [1], for Hass-Thompson-Thurston’s work on common stabilizations [2] and for Bachman-Cooper-White’s theorem that the radius of an embedded ball in a hyperbolic 3-manifold implies a lower bound on the Heegaard genus [3]. Minimal and harmonic surfaces are in some sense very different, but they both can be used (in different contexts) for the same purpose: to find a representative for a homotopy/isotopy class of surfaces in which the sectional curvature is bounded. By the Gauss-Bonnet theorem, this implies that the area of the surface is bounded above by a constant times its genus, and this gives us the necessary link between the topology and the geometry.

Given a separating surface in a geometric 3-manifold , consider the area of the surface divided by the volume of the smaller of the two complementary components. The Cheeger constant of is the infimum of this value taken over all separating surfaces in . Lackenby’s work uses this idea to compare the Heegaard genus of a 3-manifold to its Cheeger constant. In particular, he uses work of Pitts-Rubinstein to find a sweep-out for in which the largest area surface is a minimal surface, and thus has the necessary curvature bound. Since this surface has maximal area of all the sweep-out surfaces, any other surface in the sweep-out has area bounded by the genus of the Heegaard surface. One of the sweep-out surfaces cuts the volume in half, so the Cheeger constant is bounded above by a constant times the Heegaard genus divided by the volume of the 3-manifold.

So the intuition that comes from Lackenby’s picture is that a low genus Heegaard surface cutting the 3-manifold in half topologically translates to a low area surface cutting it in half geometrically. (Ok, this is kind of a stretch, but bear with me.) If we have a geometric picture of a 3-manifold and we want to look for irreducible Heegaard splittings, we should look for small area surfaces that cut the manifold into large volume pieces. What kind of geometric picture am I talking about? In a guest post on this blog, Ian Biringer discussed his recent work with Juan Souto showing that all non-Haken, hyperbolic 3-manifolds with injectivity radius bounded from below and rank bounded from above are built from pieces pulled from a finite list, glued together along their boundary components. (They conjecture that the same result is true for Haken 3-manifolds as well.) Brock, Minsky, Namazi and Souto have been working on explicitly constructing metrics on such manifolds.

The geometry of one of these 3-manifolds looks like the cores of the finite pieces, with long surface-cross-interval tubes connecting them. As the gluing maps connecting the pieces become more complicated, the surface-cross-interval tubes become longer, though the area of the surface cross sections stay bounded. Thus for sufficiently complicated gluing maps, the most efficient way to cut such a 3-manifold in half (geometrically) should be to cut across the surface cross sections of the tubes. A Heegaard surface that behaves in this way will probably be what’s called an amalgamation along those surfaces. This is the intuition that comes from the geometric picture, and while one could probably put together a geometric argument, there are already a number of purely topological proofs of results along these lines.

In fact, there is a whole collection of papers in which this idea evolved in the topological setting, independently of the geometric picture. One of the most recent (and most general) results is by Tao Li [4] (which I described here), though there is another branch of this family of results that takes a slightly different point of view as in [5]. The techniques in the topological setting prove to be much nicer than those in the geometric setting. (For an example of the techniques in the geometric setting see the recent paper by Hass, Scott and Thompson [6], which I described here.) However, the intuition from the geometric setting seems to me much better. In fact, I don’t know any way to describe the intuition behind the topological point of view other than to explain some of the proofs (and I won’t do that here). So it seems like the geometric picture should be valueable as a source of conjectures that can be proved topologically. I don’t have any good conjectures right now, but I bet they’re out there.

[5] Kobayashi, Tsuyoshi, Qiu, Ruifeng, The amalgamation of high distance Heegaard splittings is always efficient.

*Math. Ann.* 341 (2008), no. 3, 707–715.

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