Low Dimensional Topology

October 16, 2011

The reducible automorphism conjecture

Filed under: 3-manifolds,Heegaard splittings,Mapping class groups — Jesse Johnson @ 8:55 pm

Recall that the mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold $M$ that take the Heegaard surface $\Sigma$ onto itself, modulo isotopies of $M$ that keep $\Sigma$ on itself. The isotopy subgroup is the group of such maps that are isotopy trivial on $M$, when you ignore the Heegaard surface. Hyam Rubinstein and I constructed a number of examples of Heegaard splittings with non-trivial isotopy subgroups [1], but all of these groups were generated by reducible automorphisms. Before our paper, Darren Long had constructed a strongly irreducible Heegaard splitting with a pseudo-Anosov element in its mapping class group [2] but the ambient manifold is a Seifert fibered space over a sphere with five singular fibers and Heegaard splittings of Seifert fibered spaces have large mapping class groups, usually including lots of reducible maps. This raises the question: Is there a Heegaard splitting with a non-trivial isotopy subgroup consisting entirely of pseudo-Anosov maps (other than the identity)?  I will go a step further and, in the spirit of my last two posts, suggest a reckless conjecture:

The reducible automorphism conjecture: The isotopy subgroup of every Heegaard splitting is generated by reducible automorphisms.

October 5, 2011

Dehn filling and genus dropping

Filed under: 3-manifolds,Dehn surgery,Heegaard splittings,Knot theory — Jesse Johnson @ 11:00 am

A common problem in low-dimensional topology is to ask how the topology and geometry of a manifold changes if you glue a solid torus into one of its torus boundary components (also known as Dehn filling) or more generally, if you glue a handlebody into a higher genus boundary component.  One topological version of this problem is to ask how the isotopy classes of Heegaard surfaces change. Every Heegaard surface  for the unfilled manifold becomes a Heegaard surface for the filled manifold, but there may be other properly embedded non-Heegaard surfaces that also become Heegaard surfaces if you cap them off after the filling. In particular these new Heegaard surfaces may have lower genus, so the Heegaard genus of the manifold could drop after filling. The quintessential example of this is a knot complement in the 3-sphere: There are knot complements with arbitrarily high Heegaard genus, but if you Dehn fill to produce the 3-sphere, then the genus drops to zero.

Of course, for such a manifold there is exactly one filling that produces the 3-sphere and one can ask how much the genus can drop for the other fillings. There are examples where Heegaard genus drops by one for a line of slopes, and the resulting Heegaard surfaces are often called horizontal.  However, Moriah-Rubinstein [1] (and later Rieck-Sedgwick [2]) showed that there are only finitely many slopes for which the genus can drop by more than one (and only finitely many lines of slopes where it drops by one.) As far as I know there are no examples where there are two slopes for which the genus drops by more than one. So one can ask:

Question: Is there a 3-manifold $M$ with Heegaard genus $g$, a torus boundary component $T$ and two slopes on $T$ such that Dehn filling along each slope produces a 3-manifold with Heegaard genus less than or equal to $g - 2$?

September 29, 2011

The generalized Scharlemann-Tomova conjecture

Filed under: 3-manifolds,Curve complexes,Heegaard splittings — Jesse Johnson @ 6:18 am

Soon after John Hempel introduced the notion of (curve complex) distance for Heegaard splittings, Kevin Hartshorn showed that the existence of an incompressible surface implies a bound on the distance for any Heegaard splitting of the same 3-manifold. Scharlemann and Tomova noted that a strongly irreducible Heegaard surface behaves much like an incompressible surface, and generalized Hartshorn’s Theorem as follows: If $M$ admits a distance $d$ Heegaard surface $\Sigma$ then every other genus $g$ Heegaard surface with $2g < d$ is a stabilization of $\Sigma$. This is a great theorem and has had huge consequences for the field, but there is one thing that has always bothered me about it: It leaves open the possibility that, for example, there may a 3-manifold $M$ with a genus three, distance 100 Heegaard surface and a second Heegaard surface of genus 201 that is unrelated to the first one. This has always seemed very unlikely to me, so I propose the following conjecture:

The generalized Scharlemann-Tomova conjecture: For every genus $g$, there is a constant $K_g$ such that if $\Sigma \subset M$ is a genus $g$, distance $d \geq K_g$ Heegaard surface then every Heegaard surface for $M$ is a stabilization of $\Sigma$.

July 25, 2011

Cut Stabilization

Filed under: 3-manifolds,Heegaard splittings,Knot theory,Thin position — Jesse Johnson @ 8:10 am

A few months ago, I wrote about a generalization of thin position for bridge surfaces of knots and links (introduced by Maggy Tomova [1]) that comes from considering cut disks, i.e. compression disks that intersect the link in a single point in their interior. One implication of this new definition is that it suggests a generalized concept of stabilization. A stabilization of a Heegaard surface is defined by a pair of compressing disks on opposite sides of the Heegaard surface whose boundaries intersect in a single point. Compressing along either of these disks produces a new Heegaard surface with lower genus. A cut stabilization occurs when one of those disks is a cut disks, and this situation comes up, for example, in Taylor and Tomova’s version of the Casson-Gordon Theorem [2]. As I will describe below the fold, cut compressions have very interesting implications for studying bridge surfaces of different genera.

July 1, 2011

Rank vs. genus: An example

Filed under: 3-manifolds,Heegaard splittings,Hyperbolic geometry — Jesse Johnson @ 2:45 pm

A Heegaard splitting for a closed 3-manifold $M$ determines a presentation for the fundamental group of $M$ in which the number of generators is equal to the genus of the Heegaard splitting. This implies that the Heegaard genus of any 3-manifold $M$ is greater than or equal to its rank (i.e. the minimal number of elements in a generating set for its fundamental group.) The reverse inequality, however, does not hold: There are Seifert fibered spaces (discovered by Boileau and Zieschang [1]) whose ranks are one less than their genera. These were generalized by Scultens and Weidman [2] to graph manifolds in which the difference (genus – rank) is arbitrarily large, though the ratio (genus/rank) is bounded. Until recently, it was unknown if a rank/genus gap could occur in Hyperbolic manifolds. However, Tao Li has just posted a preprint [3] where he constructs hyperbolic examples in which the difference between rank and genus (though not the ratio) is arbitrarily large. This makes two existing questions much more intriguing: Is there a good characterization of hyperbolic manifolds with a rank/genus gap?  Is there a bound on the ratio rank/genus?

Note: You may have seen a preprint a few years ago claiming to prove that rank/genus could be arbitrarily small. However, that paper turned out to rely on incorrect results from another paper, as I mentioned in an earlier post.

[1] Boileau, M., Zieschang, H., Heegaard genus of closed orientable Seifert 3-manifolds. Invent. Math. 76 (1984), no. 3, 455–468.

[2] Schultens, Jennifer, Weidman, Richard, On the geometric and the algebraic rank of graph manifolds. Pacific J. Math. 231 (2007), no. 2, 481–510.

June 22, 2011

Spinning with bridge spheres

Filed under: 3-manifolds,Heegaard splittings,Knot theory — Jesse Johnson @ 3:16 pm

In a post from a couple of years ago, I mentioned a paper by Yeonhee Jang showing that there are 3-bridge knots with infinitely many distinct 3-bridge spheres.  This paper has now been published in Top. App., and you can find the link on Jang’s web page.  She recently posted to the arXiv a second paper on the subject [1], showing that every (non-split) link with infinitely many three-bridge spheres has a relatively simple form consisting of a two-bridge knot with a second one-bridge component that wraps around the two-bridge knot in a very simple way that you can see below the fold.

May 30, 2011

A funny thing about circular thin position

Filed under: 3-manifolds,Heegaard splittings,Knot theory,Thin position — Jesse Johnson @ 2:34 pm

At the AMS Sectional in Iowa City a few months ago, there were a number of talks about circular thin position (i.e. circular generalized Heegaard splittings).  This is an idea that was introduced by Fabiola Manjarrez-Gutierrez for studying knot complements [1], though as she notes, it can be applied to any 3-manifold with infinite first homology.  Alexander Coward gave a talk about using these ideas to study knots with unknotting number one (i.e. knots that become the unknot after a single crossing change) and he pointed out a difference between circular thin position and standard thin position that really blew me away: There are infinitely many circular generalized Heegaard splittings for the unknot that come from stabilizing the minimal thin position exactly once. Below the fold, I’ll give a brief description of circular thin position, then explain this surprising phenomenon.

May 5, 2011

Why cut disks?

Filed under: 3-manifolds,Heegaard splittings,Knot theory — Jesse Johnson @ 1:26 pm

Classical thin position for links is a relatively simple idea:  Draw a link diagram on a piece of paper, then think of a horizontal red line sliding from above the kink to below, keeping track of the number of points of intersection at each time that it’s transverse to the link.  To get to thin position, we want to redraw the diagram so that the number of intersections at any given point stays as small as possible.  This perspective makes it seem like thin position is about isotoping knots around, but Scharlemann and Thompson noticed that it’s actually about surfaces: If you think of those red lines as the projections of horizontal spheres, then thin position is the thinnest way to push a sphere from above the knot to below it. Each time the sphere passes through the knot is defined by a bridge disk whose boundary consists of an arc in the knot and an arc in the sphere.  If you get rid of the knot and replace those bridge disks with compressing disks, you get Scharlemann and Thompson’s thin position for 3-manifolds [1].

Hayashi and Shimokawa [2] took this a step further by defining a type of thin position in which one looks at surfaces in an arbitrary 3-manifold containing a link and considers both compressing disks and bridge disks.  (The relationship between these different types of thin positions is the basis for the axiomatic thin position that I wrote about earlier, and which is now carefully written up in a preprint.)  Maggy Tomova has generalized Hayashi-Shimokawa thin position even more (and used this type of thin position very effectively) by adding into the mix what she calls a cut disk – a disk whose boundary is in the surface and whose interior intersects the link in a single point.  It has taken me a long time to get used to and appreciate this idea, but now that I realize how perfectly cut disks fit in with the general thin position machinery, I would like to try and explain it below.

April 20, 2011

Open problems from the AMS special session on thin position

Filed under: 3-manifolds,Heegaard splittings,Knot theory — Jesse Johnson @ 2:15 pm

Back in March, I co-organized a special session on thin position with Maggy Tomova at the AMS sectional meeting in Iowa City.  We ended the session with a discussion of open problems, both directly and indirectly related to thin position.  Scott Taylor kept a list of the problems that were suggested, and TeX’d it up afterwards.  I put the PDF on my home page and you can see it here.

February 14, 2011

The good, the bad and the one-sided

Filed under: 3-manifolds,Heegaard splittings — Jesse Johnson @ 11:44 am

If you manage to embed a non-orientable surface into an orientable 3-manifold, the embedding will be what’s called one-sided:  If you choose a normal vector somewhere on the surface, then drag it by its endpoint around a non-orientable loop in the surface, it will come back on the other side.  An equivalent definition is that the boundary of a regular neighborhood of the surface will have a single boundary component.  (In many popular accounts of topology, one-sidedness seems to be the property of Mobius bands and Klein bottles that is described, since non-orientable is harder to define.)  These surfaces are notoriously misbehaved.  For example, unlike two-sided surfaces, a one-sided surface that is geometrically incompressible (no embedded disk has essential boundary in the surface) will often have a non-injective fundamental group.  However, Loretta Bartolini has found [1] that in some cases they can be understood better than their two-sided counterparts.

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