In my last two posts, I defined the complex of surfaces S(M) for a 3-manifold M – a cell complex with a partial ordering defined on its vertices. During the time between writing those posts and working on this one, I realized that I want to make one slight change in the definition of the complex of surfaces (I’ve edit the older posts to reflect this) by having a separate edge for each compressing disk in a surface. This has the side effect that the descending link of each vertex (the subset of the link of the vertex spanned by edges along which the complexity decreases) is exactly the disk complex of the surface, so the link index of a vertex (one plus the dimension of the first non-trivial homotopy group of the descending link) is precisely Dave Bachman’s topological index. The reason I made the change is that with this new definition, each edge represents a compression body in M (unique up to isotopy). Note that some pairs of vertices may have multiple edges between them.
Last time, we defined a complexity on paths in S(M) and showed that the local maxima in a slender (locally thin) path all have link index one. The process of thinning a path is very similar to the process of weak reduction in Scharlemann-Thompson thin position, but not exactly because a generic path in the complex of surfaces may not correspond to a generalized Heegaard splitting. To get things to work out correctly, we need to add some labels to S(M) and two more axioms.
Recall that vertices in S(M) are transversely oriented, strongly separating surfaces in M and edges correspond to compressing one surface and producing the other. Each compressing disk is on either the positive side of the surface or the negative side, so we will label each edge positive or negative based on whether it is defined by a compression on the positive side or the negative side. We will say that an edge path in S(M) is oriented if the complexity always increases along negative edges and decreases along positive edges. Such a path has the property that the compression bodies defined by its edges have pairwise disjoint interiors.
If a path comes from a Heegaard surface then all the edges going up to the Heegaard surface from the initial sphere have negative labels and the edges descending to the next sphere have positive labels, so this path is oriented. In fact, for every path with this property, the maximal vertex in the middle represents a Heegaard surface. A splitting path will be defined as an oriented path from the the sphere bounding a ball on the negative side to the sphere bounding a ball on the positive side. The compression bodies defined by the edges in the strictly increasing and strictly decreasing segments of the path form larger compression bodies bounded by the surfaces corresponding to the local maxima and local minima. Thus a splitting path determines a generalized Heegaard splitting for M. (Whether or not you know the definition of a generalized Heegaard splitting, you should be able to work it out the definition from the definition of a splitting path.)
The thinning operation we defined in the last post, where we pushed the path down across a number of squares, is problematic because when we thin a splitting path, we want to make sure that the resulting path is also a splitting path. As it’s defined so far, the resulting path may increase along positive edges and decrease along negative edges. To deal with this we introduce our first new axiom:
The parallel orientation axiom: Opposite edges of squares are either both positive or both negative.
The reader can check that this axiom holds for S(M). This implies that when sliding a path across a square, starting with an oriented path, the new path will be oriented if and only if two of the edges in the path are replaced by two new edges. If all four edge have the same sign then the initial two edges must go from the minimum vertex of the square to the maximum, as do the new two edges. If the edges of the square have two different signs then the initial edges are both adjacent to the maximum and the new edges are adjacent to the minimum (or vice versa). The first type of move (which I’ll call a horizontal move) does not change the complexity of the path, while the second move (which I’ll call a vertical move) always increases or decreases it.
We should also add another move that I didn mention in the previous posts: If we ever see two consecutive edges of a path that form a loop in S(M) then we can remove these two edges. We may also occasionally be able to add such a pair of edges. The enterprising reader can check that these moves correspond to stabilizing or destabilizing part of the generalized Heegaard splitting.
We will say that two oriented paths in S(M) are equivalent if they are related by a sequence of horizontal moves. These moves do not affect the maximal and minimal vertices in the path. It turns out that two splitting paths are equivalent if and only the corresponding generalized Heegaard splittings are isotopic in M. (This is a non-trivial fact, but too technical to write up here.) If we look at a maximum in a splitting path and consider all the equivalent paths, the possible edges coming into the vertex determine two subsets of the descending link: One spanned by all the negative edges and one spanned by the positive edges. In other words, a vertex of the descending link is in one of these sets if the corresponding edge can be extended to a path that can be pushed into the original path by horizontal face slides.
I’ll call these two subsets of the descending link the positive and negative links. Note that they are determined not just by the vertex, but by the path that goes through it. This is important. Different paths through a given vertex may determine different positive and negative links and their union may not be the whole descending link. In particular, each of the paths descending from the vertex defines a compression body. An edge in the descending link will be in the positive or negative link if and only if the corresponding disk can be isotoped into one of these compression bodies.
We will say that a maximal vertex in a given path is weakly reducible if the union of the positive and negative links is connected. Otherwise, it is strongly irreducible. Again, this condition depends on the oriented path, not just the vertex. If a maximum in a splitting path is weakly reducible then we can choose an equivalent path in which the last edge before the maximum and the first edge after the maximum are adjacent edges in a square in S(M). There is then a vertical move that reduces the complexity. The reader familiar with thin position for 3-manifolds can check that this vertical move corresponds to weak reduction of the generalized Heegaard splitting.
We thus conclude that a slender splitting path has strongly irreducible maxima. Traditional thin position for 3-manifolds suggests that a thin generalized Heegaard splitting should not only have strongly irreducible maxima, but incompressible minima. The proof in the usual setting relies on a lemma of Casson and Gordon and we will encode this lemma as an axiom:
The Casson-Gordon axiom: If a minimum vertex in a splitting path has a positive edge descending from it then either the following maximum is weakly reducible or the following minimum has a positive edge descending from it. Similarly, if a minimum vertex has a negative edge descending from it then either the previous maximum is weakly reducible or the previous minimum has a negative edge descending from it.
This axiom for S(M) follows from Casson and Gordon’s lemma. Given a slender splitting path, with all maxmima strongly irreducible, we’ll assume for contradiction there is a local minimum with a descending edge. If this edge is negative then by the Casson-Gordon axiom, the previous minimum must have a descending negative edge as well. We repeat the argument until we get to the first minimum in the path, which can’t have a descending edge because it’s a sphere. The same argument gets rid of positive descending edges, so we conclude that a slender path representing a generalized Heegaard splitting has strongly irreducible maxima and incompressible minima. So we’ve recovered the standard thin position for 3-manifolds from the complex of surfaces.