In this entry I want to discuss the ordering on sequences of generalized Heegaard splittings that Dave Bachman has suggested in his (potential?) proof of the Gordon conjecture. We will start by defining an ordering on multi-component closed surfaces. The ordering I want to suggest is not the simplest, but it begins a pattern that will appear later. Given a not necessarily connected, closed surface, we will first arrange its components in non-decreasing order by genus. To compare two such surfaces, we will apply a lexicographic ordering: Compare the first/largest components of each and if they’re different, the surface with the larger first component is larger. Otherwise, compare the second largest components and so on. This ordering has the nice property that if you add a handle to a component of a surface, or between two non-sphere components of a surface, you (strictly) increase its complexity. Conversely, if you compress a surface, you (strictly) decrease its complexity.
Recall from last time that a generalized Heegaard splitting is a sequence of surfaces in a 3-manifold that cut the 3-manifold into compression bodies with certain conditions. A generalized Heegaard splitting is induced by a Morse function on the 3-manifold (whose critical points are not necessarily ordered by index) as follows: Given a Morse function, consider representatives of the regular level sets between its critical points. At an index zero or one critical point, the complexity of the level set increases. At an index two or three or critical point, the complexity decreases. (There are two trivial exceptions to this that I will ignore.) A generalized Heegaard splitting is the union of all the locally maximal representatives (called thick levels) and the locally minimal representatives (the thin levels) of level surfaces.
To compare two generalized Heegaard splittings, we will apply a lexicographic ordering to the thick levels. In other words, given two generalized Heegaard splittings, we first compare the most complex thick level in each splitting, then the second most complex and so on. Just as the ordering for surfaces is compatible with the operation of compressing on surfaces, this ordering on generalized Heegaard splittings is compatible with two operations. In the Morse function, these operations correspond to moving two critical points past each other (i.e. changing the order of their levels) and canceling (or uncanceling) pairs of critical points whose indices differ by one.
A sequence of generalized Heegaard splittings (SOG) is one in which consecutive generalized splittings are
related by these two moves. Just as we defined an ordering on generalized Heegaard splittings using our ordering on surfaces, Bachman suggests defining an ordering on SOGs by applying a lexicographic ordering to the locally maximal generalized Heegaard splittings in the sequence.
The ordering on generalized Heegaard splitting (which was defined by Scharlemann and Thompson as an adaptation of Gabai’s definition of thin position for links) is useful because under this complexity and the two moves mention above, locally minimal generalized Heegaard splittings have the property that their thin levels are incompressible and their thick levels are strongly irreducible Heegaard splittings of the complements of the thin levels. Scharlemann and Thompson state the result for absolutely minimal generalized splittings, but the proof works just as well for locally minimal splittings. Bachman has suggested that there should be an analogous result for minimal SOGs, which I will discuss next time.