I want to continue my series of posts about axiomatic thin position by considering what it means for a path in the complex of surfaces S(M) (or any complex satisfying the same axioms) to represent an index one vertex in the complex of paths in S. (If you’re just tuning in now, you may want to first look at parts one, two, three and four.) Recall from the last post that a vertex in P(S) represents an equivalence class of oriented paths in S, i.e. a path in S, modulo face slides that do not change the complexity of the path. Edges in P(S) correspond to weak reductions/vertical face slides of paths in S. Faces in P(S) come from pairs of weak reductions that commute with each other. (Since this is a blog post, I will be vague about what that last sentence means.)
The complex of paths P(S) satisfies the three axioms that I defined in my second post so when we consider slender paths in this complex, we will be interested in index-zero and index-one vertices. Last time, I pointed out that index zero vertices in P(S) correspond to slender paths in S and that a path in S corresponding to an index-one vertex in P(S) will have exactly one weakly reducible maximum and the rest will be strongly irreducible. Today, I’d like to determine what we can say about the one weakly reducible maximum.
Let v be the one weakly reducible maximum in an oriented edge path E that determines an index one vertex in P(S). Recall that two weeks ago, we defined two subsets of the descending link of v: the positive and negative links, determined by all the edges before and after v in the different paths equivalent to E. Each of these sets is connected and a weak reduction of a path equivalent to E determines an edge between the two sets. That means there is a one-to-one correspondence between edges in P(S) descending from the vertex representing E and edges in the descending link of v that connect the two sets. If the descending link of the vertex representing E has index-one then it is disconnected. The different components of its descending link determine disjoint sets of edges connecting the positive and negative links of v, so the subcomplex of the descending link spanned by the positive and negative links is not simply connected.
Motivated by this fact, we will say that a maximum v in a path E is critical if the subcomplex spanned by the positive and negative links is connected but not simply connected. (Note that this is closely related to index-2, except that we are only considering a subset of the descending link of v.) The above discussion implies that if a path in S represents an index-one vertex in P(S) then it has one critical maximum and the rest of the maxima are strongly irreducible.
Along with the discussion of strongly irreducible vertices, I defined the Casson-Gordon axiom, which implies that if all the maxima in a path are strongly irreducible then the minima are index-zero. It turns out that a similar axiom holds for critical maxima:
The Bachman axiom: Let v be a maximum in an oriented path in a height complex S. Let v-, v+ be the minima of the path before and after v, respectively. If v– has a positive edge descending from it and v is critical then there is a positive edge descending from v+. If v+ has a negative edge descending from it and v is critical then there is a negative edge descending from v-.
This might more fittingly be called the index-two Casson-Gordon axiom, and there is a fairly straightforward way to generalize both the statement and hte proof to higher index, but I only need these two axioms. By applying this axiom as we did the Casson-Gordon axiom, one can show that a path representing an index-one vertex in P(S) must have index-zero minima.
The Casson-Gordon axiom comes from the following corollary of a Lemma of Casson and Gordon: Given a 3-manifold M such that there is a compressing disk D for the boundary of M, then every Heegaard splitting of M is weakly reducible. (Originally, Casson and Gordon showed that any Heegaard surface can be isotoped to intersect some boundary complessing disk in a single loop, generalizing a parallel result of Haken for an essential sphere rather than a disk.) The surfaces corresponding to v-, v, and v+ in the axiom determine a submanifold of M and a Heegaard surface for this submanifold. A positive edge descending from v– determines a compressing disk. If this disk crashes through v+ then it determines a positive edge descending from v+. Otherwise, it determines a boundary compressing disk for the submanifold, and the lemma implies that v is weakly reducible.
In fact, the corollary can be proved without citing Casson and Gordon’s lemma. The idea is that every Heegaard surface for M must intersect the disk D and this intersection must contain a loop that bounds a compressing disk for the Heegaard surface (because every loop of intersection bounds a disk in D). Given any Heegaard surface, we can isotope it so that the intersection contains the boundary of a disk on one side, then isotope it so that the intersection contains the boundary of a disk on the other side. During the second isotopy, the intersection of D with every intermediate surface contains the boundary of some compressing disk, on one side or the other. Somewhere it has to switch from one side to the other, and when it makes that switch, one can show that the disks on opposite sides are disjoint. (I think this proof was originally devised by Hyam Rubinstein, but if any reader knows better, please let me know.)
What makes this proof work is having a surface that must intersect every Heegaard surface for M and such that all loops in this surface bound disks. So this proof would work just as well for an essential sphere in M. Also, the proof uses a one dimensional family of surfaces isotopic to a given Heegaard surface, but we can adapt it to a larger family of surfaces, say a 2-dimensional family determined by a loop in the disk complex for the surface. By doing this, one can prove that given a 3-manifold with either an essential sphere or a compressing disk for its boundary, every Heegaard splitting has simply connected disk complex. If this Heegaard surface comes from a maximum in a path, then this implies it is not critical, giving us the Bachman axiom.
If we apply the sphere case of this argument, we get Dave Bachman’s proof of the Gordon conjecture. Recall that the Gordon conjecture states that a connect sum of two unstabilized Heegaard splittings is unstabilized. Each of the original Heegaard splittings determines a path in the complex of surfaces for that 3-manifold with a single maximum. To take the connect sum of the Heegaard splittings, we have to project the paths into the complex of surfaces for the connect-sum manifold and then “concatenate” the paths. (Again, I’m leaving out lots of details.) If the resulting Heegaard splitting is stabilized then we can slide it around to get a path where the stabilization shows up as a length two loop in the path.
This determines a path in the complex of paths P(S), so we might as well pick a slender path where all the maxima are index-one and the minima are index-zero. Each minimum or maximum of the path in P(S) is a path in S and we will let E be one of these edge paths in S. The maxima of E determine Heegaard surfaces for submanifolds bounded by the minima of E. Because the minima of E are index-zero, the essential sphere that comes from the connect sum must be isotopic into the complement of these incompressible surfaces. Because the complementary submanifolds have either strongly irreducible or critical Heegaard surfaces (given by the maxima in E) the sphere that comes from the connect sum cannot be essential in any of the submanifolds so it must be parallel into one of the incompressible surfaces. Thus the connect sum sphere is present in a minimal surface of every path E. This means that every intermediate path in S from the original path to the obviously stabilized path is a concatenation of two paths in the original manifolds. If a stabilization appears in on one of these path, it must appear in one of the original Heegaard splittings, implying that one of the original splittings was stabilized.