My goal in this post is to begin defining a system of axioms for thin position that can be applied in a variety of different contexts. In my last post, I defined the complex of surfaces S(M) for a 3-manifold M, a cell complex with a partial ordering on its vertices. It turns out that Scharlamann-Thompson thin position can be defined entirely in terms of this cell complex and its ordering, then interpreted topologically. So one can take this as a model for defining other types of thin positions given a cell complex and an ordering that meet the appropriate axioms.
For now, though, I want to explain how the link index I defined last time (which is closely related to Bachman’s topological index) comes out of the discussion of thin position. For this, we’ll need two axioms that one can readily check hold for the surface complex (I”ll define more axioms in the next post):
Strictness axiom: If two vertices are connected by an edge then one has strictly greater complexity than the other.
Parallel axiom: Every square has a maximum vertex and a minimum vertex and these are diagonally opposite each other.
(I call the second one the parallel axiom because it implies that if you put an arrow on each edge pointing down then parallel sides of each square will point in the same direction. Note that you can extend this to a characterization of maxima and minima of higher dimensional cubes.)
The vertices of S(M) are isotopy classes of transversely oriented, separating surfaces in M. There are two special surfaces that we can single out: There is a sphere that bounds a ball on its positive side and sphere that bounds a ball on its negative side. (These are really the same surface, but with opposite orientations.) The corresponding vertices of S(M) I’ll call v+ and v-, respectively. Given an oriented Heegaard surface in M, there is a sequence of compressing disk on either side such that compressing along one set produces v+ and along the other set produces v-. Thus a vertex in S(M) representing a Heegaard surface is characterized by an edge path to v+ along which the complexity strictly decreases and a similar edge path to v-. But for a moment, lets forget about Heegaard surfaces and just look at arbitrary edge paths in S(M).
We can define an ordering on edge paths in S(M) as follows: A vertex in an edge path will be called a local maximum if its complexity is greater than the vertex before it and the vertex after it in the path. The complexity of a path will be the n-tuple of complexities of the local maxima in the path, arranged in non-increasing order. To compare two edge paths, we simply apply lexicographic ordering to the n-tuples defined by the local maxima. A thin path between two vertices is one that has minimal complexity over all paths between those vertices. (Note: now we can define a complex of edge paths and apply thin position to that complex, but that’ll have to wait for a future post…)
Given a path, we can sometimes find a thinner path by sliding two of the edges across a square cell in S(M). A path will be called slender if one cannot reduce its complexity by sliding, without first increasing its complexity. (So a slender path is “locally thin”.) Note that a thin path is slender, but a slender path may not be thin. If we look at a local maximum of a thin path, the edges coming into the vertex and leaving the vertex are in the descending link (discussed last time). If they are in the same component of the descending link, then the path connecting them in the descending link determines a sequence of squares sharing the vertex. By the parallel axiom, this vertex is maximal in each square, so if we push the path across all these squares, we get a (possibly much longer) path in which we replace the one local maximum with a sequence of local maxima with strictly lower complexity. Because of the lexicographic ordering, this new path is thinner than the original.
Thus if we have a slender path, the descending link of each locally maximal vertex in the interior of the path must have at least two components. In other words, the dimension zero homotopy group is non-trivial so every local maximum link index one. Note that the strictness axiom implies that every path has a locally maximal vertex either in its interior or at one of its endpoints.
If you want to see how index-two surfaces come into play, consider a (possibly immersed) disk in S(M) composed of squares. A vertex in this disk is a local maximum if every edge in the disk touching that vertex descends from it. The disk thus defines a loop in the descending link of the vertex. If that link is homotopy non-trivial in the descending link then the vertex has index one or two. Otherwise, there is an immersed disk in the descending link bounded by this loop. This disk determines a collection of cubes across which we can isotope the original disk to replace the locally maximal vertex with a collection of lower maxima.
Now, it’s important to note that what we’ve proved so far about thin paths in S(M) doesn’t tell us anything about Heegaard splittings, since not every path in S(M) corresponds to a generalized Heegaard splitting. If we start with a path coming from a Heegaard surface and try to thin it, we may end up with a path that doesn’t represent a Heegaard surface. In order to prove the statements that one expects from Scharlemann-Thompson thin position, we need to add in a few more axioms. But I’ll save that for the next post.
Addendum: I realized the morning after posting this that I need one more axiom to make sure that the process of reducing the maxima eventually ends:
Minima axiom: Any strictly decreasing edge path is finite.
The reader can check that this implies we can always find a slender path by pushing down the maxima in a path.