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Cameron McA Gordon (Editor)
Abstract

These are problems on Heegaard splittings, that were raised at the Workshop, listed according to their contributors: David Bachman, Mario Eudave-Muñoz, John Hempel, Tao Li, Yair Minsky, Yoav Moriah and Richard Weidmann. In [34] Hyam Rubinstein gives a personal collection of problems on 3–manifolds.

Problems]Heegaard splittings of 3–manifolds (Haifa 2005)
Problems

1 David Bachman

We analyze what is known, and what is not known, about the following question:

TheoremQuestion 1.1   ()

Which manifolds have infinitely many Heegaard splittings of the same genus, and how are the splittings constructed?

The only known examples of 3–manifolds that admit infinitely many Heegaard splittings of the same genus are given in Sakuma [35], Morimoto and Sakuma [28] and Bachman and Derby-Talbot [3]. By Li’s proof of the Waldhausen conjecture [19], any such manifold must have an essential torus . In the presence of such a torus, it is easy to see how an infinite number of Heegaard splittings of the same genus might arise. Simply take any splitting that intersects , and Dehn twist the splitting about . But is this the only way such an infinite collection might arise? What can be said about the manifold if this construction does not produce an infinite collection of non-isotopic splittings?

We analyze what is known when the torus is separating. (Some of the results described below hold in the non-separating case as well.) Suppose then is an essential torus which separates a closed, orientable, irreducible 3–manifold into and . Then we may think of as being constructed from and by gluing by some homeomorphism, .

Fix separate triangulations of and (these do not have to agree in any way on ). By a result of Jaco and Sedgwick [13] the sets and of slopes on and that bound normal or almost normal surfaces in and are finite (see Bachman [2] for a further discussion of the almost normal case). We may thus define the distance of to be the distance between the sets and , as measured by the path metric in the Farey graph of .

TheoremTheorem 1.1   (Bachman, Schleimer, Sedgwick [4])

If then every Heegaard splitting of is an amalgamation of Heegaard splittings of and .

The relevance of this theorem is that Dehn twisting an amalgamation about the torus will not produce non-isotopic Heegaard splittings. Hence, when the manifold can only admit an infinite collection of non-isotopic Heegaard splittings if or does. The converse, however, is more subtle. That is, if we know or has infinitely many non-isotopic Heegaard splittings of the same genus, then does it follow that does as well? We conjecture the following:

TheoremConjecture 1.1   ()

If and or has infinitely many Heegaard splittings of the same genus then does as well.

When or , there is the possibility that there is a strongly irreducible Heegaard splitting of . Let denote such a splitting surface. By a classic result of Kobayashi, can be isotoped to meet in a non-empty collection of loops that are essential on both surfaces.

One way Dehn twisting about can fail to produce a non-isotopic Heegaard splitting is if is a fiber of a fibration of . Then the effect of the Dehn twist can be “undone” by pushing around the fibration. This is precisely what happens when is a separating vertical torus in a Seifert Fibered space. (For a complete resolution of Question 1.1 in the case of Seifert Fibered spaces, see [3].) A second thing that may happen is that is the union of two pages of an open book decomposition of . Then the effect of the Dehn twist can be undone by spinning about the open book decomposition. We conjecture that these are the only ways that Dehn twisting about can fail to produce non-isotopic Heegaard splittings:

TheoremConjecture 1.2   ()

If admits an essential torus , separating into and , and a strongly irreducible Heegaard splitting , then either Dehn twisting about produces an infinite collection of non-isotopic Heegaard splittings, or can be isotoped so that or is either a fiber of a fibration of or or two pages of an open book decomposition of or .

2 Mario Eudave-Muñoz

Let be a closed surface of genus standardly embedded in , that is, it bounds a solid torus on each of its sides. We say that a knot has a presentation or that it is in a position, if has been isotoped to intersect transversely in points that divide into arcs, so that the arcs in each side can be isotoped, keeping the endpoints fixed, to disjoint arcs on . The genus–1–bridge number of , , is the smallest integer for which has a –presentation. We say that a knot is a knot if . If is a –knot, then it is easy to see that has tunnel number one. On the other hand, if has tunnel number one, it seems to be very difficult to determine . It has been of interest to find tunnel number one knots with large genus–1–bridge number, see for example Kobayashi and Rieck [16].

Moriah and Rubinstein [24] showed the existence of tunnel number one knots with . Morimoto, Sakuma and Yokota also showed this, and gave explicit examples of knots , with tunnel number one and [29]. It was shown by Eudave-Muñoz [7], that many of the tunnel number one knots constructed in [5] are not –knots; this was extended in [8], where it is shown that many such knots are not –knots. We remark that such knots can be explicitly described, for example the knot shown in Figure 13 of [8] satisfies . Combining results of [6] and [9], it is also possible to give explicit examples of knots , with tunnel number one and , for example the knot shown in Figure 4 of [6]. Valdez-Sánchez and Ramírez-Losada have also shown explicit examples of tunnel number one knots with (personal communication). These knots bound punctured Klein bottles but are not contained in the –knots bounding Klein bottles determined by the same authors [33].

Johnson and Thompson [14], and independently Minsky, Moriah and Schleimer [22] have shown that for any given , there exist tunnel number one knots which are not –knots. The two papers use similar techniques, prove the existence of such knots, but do not give explicit examples. For a tunnel number one knot, let be the Heegaard splitting of the knot exterior determined by the unknotting tunnel, and let denote the Hempel distance of the splitting [12]. In [14] and [22] the existence of tunnel number one knots with large Hempel distance is shown, and then results of Scharlemann and Tomova [37], [40] are used to ensure the knots have large genus–1–bridge number.

We propose the following problems:

TheoremProblem 2.1   ()

For a given integer , give explicit examples of tunnel number one knots with .

TheoremProblem 2.2   ()

For a given integer , give explicit examples of tunnel number one knots with .

TheoremProblem 2.3   ()

For a given integer , give explicit examples of tunnel number one knots with , but with bounded Hempel distance, say with .

3 John Hempel

Questions on the curve complex

Let be a surface and be its curve complex whose vertices are the isotopy classes of essential simple closed curves in and whose –simplexes are determined by distinct vertices with pairwise disjoint representatives. The distance between vertices is the number of edges in a minimal edge path joining them. Understanding this distance function is a daunting task – it is hard to tell where one is headed as one moves away from a given vertex:

TheoremQuestion 3.1   ()

Given vertices of is there an “easy” way to find a vertex with and ?

One answer is given by K Shackleton [38], but it involves extending the curves to multicurves which satisfy an additional “tightness” condition and requires a search space whose size grows very rapidly with the complexity of the problem. Also, the complexity is measured in terms of the intersection numbers of the multicurves involved. This may not be the most natural measure and could be a distraction. I am hoping for something easier. The difficulty will be in making a proof.

If , then splits into contractible regions, each with an even number of edges, alternating between and . We assume there are no bigons. Euler characteristic calculations yield that most of these regions are squares. When we look in , a component of split along , we see families of parallel arcs from successively in the boundaries of these squares whose unions we call stacks. deforms to a graph with one vertex in each large region of and one edge crossing each –stack (in ). Choose a maximal tree in . This determines a free basis for whose elements are represented by simple closed curves each of which crosses exactly one of the –stacks corresponding to an edge of and crossing it once. We call any such basis a –determined free basis and its elements –determined free generators. They are in many ways the most simple, relative to , curves in . It is natural to ask:

TheoremQuestion 3.2   ()

Is there some –determined free generator for with ? Does every –determined free basis contain such an element?

Caution: there are examples in which not every –determined free generator, satisfies . If Question 3.2 can’t be answered, then:

TheoremQuestion 3.3   ()

In terms of word length, in a –determined free basis, how far must we look in order to find a simple closed curve with ?

4 Tao Li

Let and be two manifolds with connected boundary and a homeomorphism. If one glues to by identifying to for each , one obtains a closed manifold . We say that is an amalgamation of and .

Let () be the set of curves in that bound essential disks in . We propose the following conjecture.

TheoremConjecture 4.1   ()

There is an essential curve () in such that if the distance between and in the curve complex () is sufficiently large, then either the minimal-genus Heegaard splitting of is an amalgamation or itself is a minimal-genus Heegaard surface in which case both and are handlebodies.

Conjecture 4.1 is a generalization of two recent theorems. In the case that and are atoroidal and have incompressible boundaries, ie, , the conjecture is proved by Souto [39] and Li [20]. Note that [20] also gives an algorithm to find and .

In the case that both and are handlebodies, and can be chosen to be empty and Conjecture 4.1 follows from a recent theorem of Scharlemann and Tomova [37]. The theorem of Scharlemann and Tomova can be formulated as: if the distance between and (ie, the Hempel distance) is large, then the genus of any other Heegaard splitting must be large unless it is a stabilized copy of .

5 Yair Minsky

For a handlebody we have an inclusion of mapping class groups, . If is a Heegaard splitting we denote , and moreover let be the kernel of the map (ie, is the group of mapping classes of that are homotopic to the identity on ).

TheoremQuestion 5.1   ()

When is finite? … finitely generated? … finitely presented?

TheoremQuestion 5.2   ()

When is equal to the amalgamation ?

TheoremQuestion 5.3   ()

When is the map injective?

When and , Akbas [1] proved is finitely presented. (Finitely generated has a longer history starting with Goeritz [11] – for details see Scharlemann [36].)

Let be the set of (isotopy classes of) simple curves in which bound disks in . Let be the simple curves in that are homotopic to the identity in . Note that contains , and is invariant under .

Namazi [30] showed if the distance of the splitting () is sufficiently large then is finite. Note also the Geometrization Theorem plus Hempel [12], plus Mostow rigidity plus Gabai–Meyerhoff–Thurston [10], implies that the image of in is finite when the splitting distance is at least 3.

TheoremQuestion 5.4   ()

When is equal to the orbit ?

Remarks   When or a connected sum of ’s, with the natural splitting, equality holds trivially. If is a lens space , with , and is the torus, then they are not equal – is all of , whereas the orbit of under is strictly smaller than . One might reasonably ask if there is equality when is hyperbolic, or if the splitting distance is sufficiently large.

6 Yoav Moriah

TheoremQuestion 6.1   ()

Give an example of a weakly reducible but non-stabilized non-minimal Heegaard splitting of a closed 3–manifold.

There are plenty of examples for manifolds with strongly irreducible Heegaard splittings of arbitrarily high genus. However there are no examples of manifolds (closed or with a single boundary component) with a weakly reducible but non-stabilized non-minimal Heegaard splitting.

TheoremQuestion 6.2   ()

Prove that if a manifold has strongly irreducible Heegaard splittings of arbitrarily high genus then the Heegaard splittings are of the form where and are surfaces with perhaps not connected. (See Moriah, Schleimer and Sedgwick [25] and Li [19].) Or give a counterexample.

TheoremQuestion 6.3   ()

There are examples by Kobayashi and Rieck (see below) of a –manifold which has both a weakly reducible and strongly irreducible minimal genus Heegaard splitting. These examples are very particular. Are there other such examples of a different nature?

TheoremTheorem 6.1   (T Kobayashi and Y Rieck [18])

There are infinitely many –manifolds which have both strongly irreducible and weakly reducible Heegaard splittings of minimal genus.

Proof.

Let , and , where the trefoil knot, with even, is any –bridge link which is not the Hopf link and is the figure knot. Let and be meridians of the –bridge link , be the longitude of and be the boundary of the annulus in .

Attach and to by gluing their tori boundaries so that is mapped to and to . We obtain a closed –manifold with incompressible tori. As the genus two toroidal –manifolds are classified by Kobayashi [15] this manifold cannot have genus 2. The surface obtained from the bridge sphere union the annulus in the trefoil complement and the two genus one Seifert surfaces of the figure complement is a closed surface of genus . Hence it is a minimal genus Heegaard splitting if it is a Heegaard surface.

Let be the two components of . So on one side of we have at the first stage the solid torus , say, glued to the genus two handlebody , which is one of the two components of , along a primitive meridional annulus to obtain a genus two handlebody. This handlebody is glued in the second stage to the genus two handlebody which is a regular neighborhood of the Seifert surface along a primitive meridional annulus. So we get a genus three handlebody .

On the other side of we have glued to the genus two handlebody along a primitive meridional annulus to obtain a genus two handlebody which is then glued again along a primitive meridional annulus to a genus two handlebody which is a regular neighborhood of the Seifert surface of the figure knot complement. Thus we get a genus three handlebody and is a genus three Heegaard splitting for .

In [15, Proposition 3.1] Kobayashi proves that a Heegaard splitting of the form is always strongly irreducible if the link is not trivial or a Hopf link.

Now consider the union along the torus boundary of and . This is a manifold with a minimal genus two Heegaard splitting and with one torus boundary component. Hence when this Heegaard splitting is amalgamated with the genus two Heegaard splitting of we obtain a genus three Heegaard splitting for which is weakly reducible as it is obtained by amalgamation. ∎

TheoremConjecture 6.1   ()

Let and be prime knots in then if and only if one of has a minimal genus Heegaard splitting with primitive meridian. (See Moriah [23].)

Recently T Kobayashi and Y Rieck disproved the conjecture for knots which are not prime (see [17]). The conjecture is known by work of Morimoto for tunnel number one knots and for knots which are connected sums of two prime knots each of which is also –small. (See [23] for more references.)

TheoremConjecture 6.2   ()

Given a knot which is not –primitive then a boundary stabilization of a minimal genus Heegaard splitting of is non-stabilized.

TheoremConjecture 6.3   ()

All twisted torus knots of type which are not –primitive have a unique (minimal) genus two Heegaard splitting. (See Moriah and Sedgwick [26].)

TheoremConjecture 6.4   ()

What are the properties of meridional essential surfaces which ensure that the tunnel number degenerate? Can these surfaces be classified?

TheoremQuestion 6.4   ()

Are there knots which are not and –bridge knots so that ? (See Morimoto [27].)

TheoremQuestion 6.5   ()

Does rank equal genus for hyperbolic –manifolds?

In [21] Lustig and Moriah defined a condition on complete disks systems in a Heegaard splitting, called the double rectangle condition. It was shown that if a manifold has a Heegaard splitting which has some complete disk system which satisfies this condition then there are only finitely many such disk systems with that property. However the double rectangle condition is clearly non-“generic”. Is it possible to define some other condition which will be “generic” in some reasonable sense? The intuition is that if the Heegaard distance of the splitting is sufficiently high then some form of this might be possible.

7 Richard Weidmann

Let be a Heegaard splitting of genus . Let be the elements corresponding to a spine of .

TheoremQuestion 7.1   ()

When is a free group of rank ?

(This would be a kind of “Freiheitsatz”.) For instance, what happens with the Casson–Gordon examples of strongly irreducible splittings of a fixed of arbitrarily high genus?

Note that the freeness holds for the examples exhibited by Namazi [31] and Namazi–Souto [32].

TheoremQuestion 7.2   ()

Find (3–manifold) groups that have only finitely many irreducible Nielsen equivalence classes of generating tuples.

Here a –tuple is called irreducible if it is not Nielsen equivalent to a tuple of type . Note that free groups and free Abelian groups have this property, in fact for those groups there is precisely one irreducible Nielsen equivalence class. Note that for Heegaard splittings it has been shown by Tao Li that closed non-Haken 3–manifolds only have finitely many isotopy classes of irreducible Heegaard splittings so those groups might be potential examples.

References


University of Texas at Austin, Department of Mathematics, 1 University Station – C1200, Austin, TX 78712-0257, USA


Received:   22 October 2007

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