Here’s an interesting example that Sangbum Cho (a student of Daryl McCullough at Oklahoma) showed me: If I did things correctly, there should be a picture below of a genus to handlebody with some simple closed curves drawn in its boundary surface. The two blue loops form a Heegaard diagram for the lens space L(3,1). The red, green and orange loops are the boundaries of disks in the handlebody.
A disk in a handlebody of a Heegaard splitting is called primitive if there is a disk in the other handlebody such that the two boundary loops intersect in a single point. The thing about a primitive disk is that compressing the Heegaard surface across a primitive disk produces a new, lower genus Heegaard splitting for the same 3-manifold. (The original Heegaard splitting is a stabilization of the new one.) Notice that each of red, green and orange loops intersects one of the blue loops in a single point (and the other blue loop in possible more points).
In the curve complex, one can consider the subset consisting of boundaries of primitive disks for each of the handlebodies in a Heegaard splitting. This comes up, for example, in Cho and McCullough’s work on the tree of unknotting tunnels and Cho’s work on the Goeritz group. In the example above, the three primitive disks form a pair of pants decomposition for the surface, corresponding to a maximal (two) dimensional simplex in the curve complex.
The interesting thing is that the dimension of the set of primitive disks for genus two Heegaard splittings of lens spaces depends on the lens space. If you try to generalize the diagram above, you can find a family of lens spaces (with criteria in terms of the continued fraction expansion of p/q) that have two dimensional primitive sets. Sangbum has a nice proof (though I can’t reproduce it here) that these lens spaces are the only ones that have a two dimensional primitive set. All other lens spaces have a one dimensional primitive set.
Although the primitive set has important connections (especially for someone like me who’s obsessed with stabilization), I don’t know of any direct applications of knowing the dimension of the set. But it is pretty interesting that it can vary within a class of such similar seeming manifolds. I think it would be interesting to see what this set can look like for general 3-manifolds. For example, does every 3-manifold have a Heegaard splitting for which the primitive set has maximal dimension?