Normal surfaces and the disjoint curve property.

I’ve been meaning to write about a recent preprint of Kobayashi and Rieck [1] that improves a result of Saul Schleimer’s [2]. Saul showed that for every 3-manifold, there is a value k such that every Heegaard splitting for that manifold of genus greater than k has the disjoint curve property (i.e. Hempel distance at most 2). Schleimer’s bound is an exponential function in terms of the number of tetrahedra in a minimal triangulation for the 3-manifold. Kobayashi and Rieck have improved the bound to a linear function of the number of tetrahedra.

These results are most interesting in the context of Tao Li’s work [3] [4] on branched surfaces and Heegaard splittings. Li showed (roughly) that every 3-manifold has a finite family of Heegaard splittings such that every irreducible Heegaard splitting for the manifold comes from Haken summing a surface from this finite family with a collection of incompressible surfaces. Combinging this with Schleimer’s result implies that in an atoroidal 3-manifold there are finitely many high distance splittings. The splittings that come from repeated Haken summing (and therefore have high genus) must all have the disjoint curve property. This suggests that there is a sort of fundamental distinction between the finitely many high distance splittings and the possibly infinitley many low distance ones.

Both Schleimer’s proof and Kobayahsi-Rieck’s proof use normal surface theory. Recall that a surface in a 3-manifold is normal with respect to a given triangulation if it intersects each tetrahedron in a collection of (normal) triangles and quadrelaterals. A surface is almost normal if it intersects each tetrahedron in a collection of triangles and quadrelaterals plus its intersection with exactly one tetrahedron also contains an octagon or an annulus whose boundary loops each intersect three or four edges.

Notice that there are two types of almost normal surfaces: those with octagon pieces and those with annulus pieces. If you will allow me to descend into sheer speculation, I’d like to suggest that there should be some sort of connection between the octagon/annulus dichotomy and the high distance/low distance dichotomy.

This speculation is motivated by the fact that in a tube almost normal surface, compressing along the tube produces a normal surface that bounds a handlebody on one side. If this handlebody is not a regular neighborhood of a subcomplex of the triangulation then the normal structure on the surface induces a two dimensional spine for the handlebody that has no order one edges. This two dimensional complex is homtotopy equivalent to a graph, but it is not collapsible so it’s a higher genus version of something like the house with two rooms. In a reasonable triangulation, one would hope to be able to avoid this sort of pathological behavior. (Note that for an octagon normal surface, the induced spines on both handlebodies have order one edges, so they could easily be collapsible onto graphs.)

I don’t know enough about normal surfaces to suggest a specific conjecture or question that would sum up what I’m trying to get at. I think it’s very unlikely that every octagon normal surface has the disjoint curve property and I know that every tube almost normal surface will not have high distance. I will thus leave it as a vague suggestion that there should be a more subtle connection lurking just in the background of all this normal surface/Hempel distance business.

Primitive disks and lens spaces

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?

Sequences of generalized Heegaard splittings 4: Criticallity

So, we have a sequence of generalized Heegaard splittings that we have chosen to be minimal with respect to the lexicographic ordering on the elements of the sequence. We’re looking at the locally maximal splittings in the sequence. Last time, I explained why all but one of the thick surfaces in this generalized Heegaard splitting should be strongly irreducible, implying that we can just focus on the one weakly reducible thick surface.

It might be useful at this point to picture a graph whose vertices are all generalized Heegaard splittings and whose edges connect splittings that are related by a weak reduction or a destabilization. The splittings right before and after the local maximum are each connected to the local max. by an edge, but any other (possibly longer) edge path from one to the other must at some point pass through a more complicated splitting. Thus we can think of this graph as a mountain range in which the local maximum is the lowest mountain pass from the preceding splitting to the following splitting.

In this graph, each edge down from the mountain pass corresponds to a pair of disks in the one weakly reducible thick surface whose boundaries are either disjoint (defining a weak reduction) or isotopic (defining a destabilization). These correspond to pairs of points in the curve complex where the handlebody sets for the Heegaard splitting defined by the thick surface either intersect or are connected by an edge. We want to divide the set of downward edges (and the corresponding pairs of vertices in the curve comples) into sets such that two edges are in the same set if and only if they are connected by a path that never goes above the original mountain pass. Since we started with a local maximum, the down edges get divided into at least two such sets.

I should mention that in Dave’s exposition, he always divides the vertices into exactly two sets, and he throws in all the other loops in the handlebody sets as well, though it doesn’t matter which set they get tossed into. I think this is distracting, so I’m not going to do it.

So, we have cut the set of weak reducing pairs for the thick surface into a number of sets such that each set determines a different side of the mountain. Dave now claims that if two such sets contain a common loop then both pairs are in the same subset i.e. on the same side of the mountain. I’m pretty sure I believe this, and moreover that proving it is a reasonable exercise once one understands generalized Heegaard splittings. However, I don’t know if I can explain the proof well enough without going into more detail than I think a blog entry should contain. So, I’ll just give a quick sketch of the idea, which you might be better off skipping and trying to work it out on your own.

Let’s focus on the case when the two pairs define weak reductions and the weak reducing pairs are D_1, D_2 and D_1, D’_2. The first weak reduction corresponds to pushing the handle corresponding to D_1 past the handle corresponding to D_2. After doing this, we’ll further push D_1 past all the other handles on the same side as D_2 that we can. The resulting generalized Heegaard splitting has a thin surface that comes from compressing the original thick surface along D_1, then compressing it along D_2, then as much as possible on the same side as D_2. If D’_2 intersects this final surface then it defines a further compression. Thus by doing all possible compressions on the side opposite D_1, we’ve compressed D’_2 out as well as D_2. If we had started by compressiong along D_1 then D’_2, we would end up with the same thin surface.  Thus there is a series of weak reductions starting with D_1, D_2 and a series of weak reductions starting with D_1, D’_2 that lead to the same generalized splitting, so we didn’t need to go over the pass.

The point of all this is that we now have a description of the local maxima of our minimized SOG in terms of the curve complex of the one weakly reducible thick surface: The intersection of one of the handlebody sets with a 1-neighborhood of the other handlebody set contains more than one component. Dave calls such a thick surface critical, and we will say that a generalized Heegaard splitting is critical if it has one critical thick surface and the remaining thick surfaces are strongly irreducible. Next time I’ll discuss how this fits into the larger picture of Dave’s two proposed proofs.

Sequences of generalized Heegaard splittings, part 3: Local maxima

In my last entry on SOGs, I described a definition of complexity for the sequences suggested by Dave Bachman. The complexity comes from a lexicographic ordering on the complexities of the locally maximal generalized Heegaard splittings in the sequence, just as the measure of complexity for the splittings comes from a lexicographic ordering on their thick levels. Just as Scharlemann and Thompson derived topological information about the thick levels by minimizing the complexity for generalized splittings, Dave has suggested that we should be able to derive topological information about the generlized splittings in the SOG by minimizing its complexity. Rather than start off the statement about what Dave says a minimized SOG should look like, I want to look for ways to minimize a SOG and then at the end I’ll see if I believe Dave’s claims.

Recall that consecutive splittings in a SOG are related by either weak reduction (moving a 1-handle past a 2-handle) or destabilization (canceling a trivial 1-with a 2-handle). If you’re having trouble picturing this, think of a knot in the 3-sphere that’s in Morse position with respect to a height function on the sphere. As the level sets pass through the knot, we will count the number of points of intersection at the local maxima. We can define a complexity for Morse positions of the knot by a lexicographic ordering on the number of intersections. We can reduce this complexity by pushing a minimum up past a disjoint maximum (a weak reduction) or canceling an adjacent maximum and minimum (destabilization). Most (though not necessarily all) of what I’m about to write works just as well in this situation.

Consider a locally maximal generalized Heegaard splitting G_1 in a minimal SOG. Right before it we have a generalized splitting G_0 and right after it we have G_2. If we can find a SOG from G_0 to G_2 where each intermediate step has complexity strictly lower than that of G_1 then we can replace the sequence G_0, G_1, G_2 by this other sequence. The new sequence might be longer, but because of the lexicographic ordering it will be counted as less complex.

We get each of G_0, G_2 from G_1 by reducing a thick surface in G_1. (To save space, I’m going to use the term reduce to mean a weak reduction or a destabilization.) If the reductions are in distinct thick surfaces of G_1, then starting from G_0, we can first perform the reduction that gets us from G_1 to G_2 (since the disks that allow us to do this reduction exist in G_0 as well) and then undo the reduction that originally got us from G_0 to G_1, but now gets us to G_2. This new intermediate splitting has complexity strictly less than G_1, so original SOG was not minimal.

Thus the moves right before and after G_1 must take place in the same thick surface of G_1. In fact, if there was a way to reduce one of the other thick surfaces in G_1, we could do this reduction, then undo the reduction that gets us from G_0 to G_1, then perform the reduction that gets us to G_2, then undo the reduction that we just added. This new path is longer, but the complexity is lower because the complexities of the intermediate splittings are lower. Thus in C_1, all but one of the thick surfaces can’t be reduced - i.e. they’re strongly irreducible.

So there’s the first important property of our local maxima - all but one of the thick surfaces is strongly irreducible. Of course, that on it’s own is not terribly useful - it could be a Heegaard splitting with exactly one thick level. Thus we need to analyze the one thick level that is weakly reducible. However, this entry is already on the long side, so I’ll leave that for next time.

Topologically non-trivial highways

I’ve been too busy these last two weeks, visiting Oklahoma and then hosting Ken Baker here at Yale, to continue my discussion of Dave Bachman’s SOGs. I’ll try to get back into that next week. For now, I wanted to point out a link to the architecture blog BLDGBLOG about a highway underpass that exhibits non-trivial braiding. Unfortunately, it seems they’re about to tear the thing down and replace it with a more reasonable (and topologically trivial) underpass.

Follow up to More on mapping class groups

I now have an answer to the question on mapping class groups of Heegaard splitting I asked a few weeks ago. It appears that for any Heegaard splitting of the 3-torus, the exact sequence from the kernel to the MCG of the Heegaard splitting to the MCG of the 3-manifold does not split (i.e. there’s no homomorphism back from the MCG of the 3-manifold). In fact, there is no injection from the mapping class group of the 3-torus to and surface mapping class group by a Theorem of Farb and Masur [1]. (Thanks go to Yair Minsky for telling me where to find the result.) Specifically, they prove that any homomorphism from an irreducible lattice in a semi-simple Lie group of rank at least two into a mapping class group has finite image. The mapping class group of the 3-torus is SL(3,Z), which is of this form, so such a homomorphism cannot be an injection.

Sequences of generalized Heegaard splittings, part 2: Complexity

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.

Sequences of generalized Heegaard splittings, part 1

In a recent post, I mentioned a paper by David Bachman [1] claiming to prove Gordon’s conjecture that a connect sum of unstabilized Heegaard splittings is unstabilized. Unfortunately this paper is a bit hard to read and certain topologists have suggested that parts of the paper seem far to optimistic. Thus in its current form the proof has not been verified, though I understand Dave is currently working with a dedicated referee to sew up all the loose ends. Regardless of the status of the proof, the main tool (sequences of generalized Heegaard splittings or SOGs) are interesting and potentially useful in other contexts. (They also come up in Dave’s examples related to the stabilization problem, which he’s currently writing up.) So I wanted to discuss what I understand about SOGs in order to see how much of Dave’s proof I believe and maybe look for other applications of SOGs. As the title suggests, I plan on writing a few entries along these lines.

In this post I will discuss the rough ideas and motivations (as I understand them) for the proof. In later entries I’ll try to flesh out the details. I want to make it clear that I am not the referee for this paper. I’m only trying to understand it in order to see if the ideas in the proof might be useful for other things.

We should start a few decades before Dave’s definition of SOGs with the paper “Reducing Heegaard splittings” by Casson and Gordon, which showed that every weakly reducible Heegaard splitting is either reducible or can be compressed (on both sides) down to an incompressible surface. Scharlemann and Thompson noticed that Casson and Gordon’s proof can be reinterpreted through an analogy to bridge position and thin position for knots. Though I’m not sure if they had this in mind when writing their paper, their notion for thin position for 3-manifolds can be derived from Morse functions.

Recall that a Morse function is a smooth function such that all the singular points are non-degenerate and are in distinct levels. For a 3-manifold, the critical points have index 0, 1, 2 or 3. At an index 0 and 1 critical point the complexity of the level surfaces increases, while at the index 2 and 3 critical point, the complexity decreases. (I won’t define complexity precisely, but you can look it up or come up with an appropriate definition as an exercise.) If we take a sequence of regular levels that are local maxima and minima (in terms of complexity) then these cut the 3-manifold into compression bodies in a very precise way that determines the structure that Scharlemann and Thompson defined, and which has come to be known as a generalized Heegaard splitting.

The set of Morse functions has a very nice structure as a subset of the (vector) space of smooth functions. The space of Morse functions is not connected, but if you throw in near-Morse functions, those where there is a single degenerate critical point, or exactly two critical points in the same level, then the space becomes connected. A path that passes through near-Morse functions corresponds to a sequence of isotopies and moves in which one of two things happen: two critical points may pass each other, or two critical points whose indices differ by one may cancel with each other or may be produced (uncancelled?). On a 3-manifold, one can work out what these two moves do to the generalized Heegaard splitting induced by the Morse function. A sequence of generalized Heegaard splittings is a sequence in which consecutive generalized Heegaard splittings are related by such moves.

One defines a complexity on generalized Heegaard splittings by a lexicographic ordering on the thick surfaces (the surfaces whose complexities are local maxima). In this ordering, one first compares the largest surfaces in each generalized splitting, then the second most complex in each, and so on. With this ordering, a generalized splittings with more surfaces of lower complexity will be less complex than one with fewer, higher complexity surfaces. Scharlemann and Thompson showed that given a minimal complexity generalized Heegaard splitting, the thin surfaces (those whose complexities are local minima) are incompressible. (Also, the thick surfaces are strongly irreducible Heegaard splittings for the complements of the thin surfaces.)

Dave’s idea is to now apply a lexicographic ordering to all the sequences of generalized Heegaard splittings between two fixed generalized Heegaard splittings and choose the path of minimal complexity. Remember, this ordering favors paths that pass through less complex generalized splittings, even if these paths are longer. The claim is that just as the minimally complex generalized Heegaard splittings picked out topological features (incompressible surfaces), the minimal complexity SOGs will do this. More precisely, there should be some feature that is preserved at each step in a minimal SOG, implying that this feature is the same in the initial and final splitting. I’ll be more precise about what is meant by this in my next entry.

Loops bounding incompressible surfaces in handlebodies

Here’s an interesting question that may not be too hard, but I don’t know the answer to it: Is there a simple closed curve in the boundary of some handlebody that is the boundary of two distinct (i.e. non-isotopic) incompressible surfaces in the handlebody? This is a side issue in a paper I wrote with Terk Patel [1] about the set of non-separating loops that bound incompressible surfaces in handlebodies, as a subset of a curve complex. (This set turns out to be 2-dense, but is still sparse enough that every Heegaard splitting of a non-Haken 3-manifold determines a pair of disjoint such sets.)

No loop can bound both a disk and an incompressible (higher genus) surface because if it did, then the disk would imply a compressing disk for the surface. However, there doesn’t seem to be an simple reason to expect a loop can’t bound two incompressible surfaces of different genus above zero.

Addendum: Here’s a construction suggested by Richard Kent (see the comments below). Take a separating, loop in the boundary of a handlebody. This loop bounds two boundary parallel surfaces, each of which is parallel to one of the complementary components in the boundary. Compress these surfaces down as much as possible. In many cases, the resulting incompressible surfaces will not be isotopic.

If you start with a non-separating loop, you can take a single boundary parallel surface whose boundary is two loops parallel to the original loop.  When you compress it down, you then have to compress along a separating disk at some point, so that you the result is two surfaces, each with one boundary component parallel to the original loop.  (Note: the construction described here sounds different than the one described by Richard, but it yields the same surfaces, and as Saul noted, in Richard’s construction you get the separating and non-separating loops without having to do two cases.)

In light of this construction, I want to reformulate the question: Is there a loop in the boundary of a handlebody bounding two distinct incompressible surfaces that don’t both come from compressing boundary parallel surfaces?

Group theory, as told by a Heegaard splitter, Part 3

Ok, one more post about group theory and then I’ll drop it. Ben Webster asked a question on my last mapping class group post about whether you could think about mapping class groups of Heegaard splittings in terms of actions on the fundamental group of the 3-manifold. Well it happens that almost any question about Heegaard splittings can be studied in terms of fundamental groups without even knowing what a Heegaard splitting is, though you need to look at more than just the fundamental group of the 3-manifold.

Stallings [1] introduced the idea of a splitting homomorphism as follows: Consider two homomorphisms from a genus g surface fundamental group onto two rank g free groups. Consider the product of the kernels of these two homomorphisms in the surface group. The product is a normal subgroup so we can take the quotient G of the surface group by this product. There are induced homomorphisms from the two free groups into G, making a commutative diamond.

Jaco showed [2] that for any two such homomorphisms, there is a way of identifying a genus g surface with the boundaries of two handlebodies, forming a 3-manifold and a Heegaard splitting for that 3-manifold, such that the inclusion maps between the fundamental groups are the two homomorphisms we started with. It follows that G is the fundamental group of the 3-manifold. Given two splitting homomorphisms (i.e. commutative diamonds of this form), we can consider a collection of four homomorphisms between corresponding elements forming a commutative cube. Such a cube corresponds to a homeomorphism between the two 3-manifolds that takes one Heegaard splitting to the other. Thus two splitting homomorphisms form a commutative cube if and only if they correspond to homeomorphic Heegaard splittings of homeomorphic 3-manifolds.

One might want to determine whether two Heegaard splittings are isotopic rather than just homeomorphic. In this case, one can consider two splitting diagrams that share the same bottom group (i.e. the fundamental group of the 3-manifold). This is sort of a commutative pair of glasses. If there are homomorphisms between the three remaining pairs of corresponding groups that make a commutative diagram (a squished cube?) then the corresponding Heegaard splittings are (in most cases) isotopic. (By “most cases” I mean for the large class of 3-manifolds for which an isomorphism between fundamental groups corresponds to a unique homeomorphism.)

So this gives one a completely algebraic way to study homeomorphism classes and isotopy classes of Heegaard splittings. The mapping class group of a Heegaard splitting is the set of commutative cubes between a fixed splitting homomorphism and itself. (Multiplication and inverses are easy to work out.) Jaco describes in [3] a construction on the commutative diamond that corresponds to stabilizing the Heegaard splitting.

Now that I’ve described all this, I should point out that it’s probably not a very useful approach to Heegaard splittings. As far as I know there are only four or five papers in existence on splitting diagrams, and though things have been proved about them, nothing has been proved using them. But who knows, maybe someone who knows more about group theory than the average Heegaard splitter could find something new.

[1] Stallings, John. How not to prove the Poincare conjecture. Topology Seminar, Wisconsin, 1965, Ann. of Math. Studies, No. 60 (Princeton 1966).

[2] Jaco, William. Heegaard splittings and splitting homomorphisms. Trans. Amer. Math. Soc. 144 1969 365–379.

[3] Jaco, William. Stable equivalence of splitting homomorphisms. 1970 Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) pp. 153–156 Markham, Chicago, Ill.