ICERM Fall 2013: Topology, geometry, and dynamics

I’ve mentioned before that the fall semester program at ICERM for 2013 will focus on computation in low-dimensional topology, geometry, and dynamics.   You can now apply to be a long-term visitor for this as a graduate student, postdoc, or other.   The deadline for the postdoctoral positions is January 14, 2013; the early deadline for everyone else is December 1, 2012 and the second deadline March 15, 2013.

There will also be three week-long workshops associated with this, so mark your calendars for these exciting events:

1. Exotic Geometric Structures. September 15-20, 2013.
2. Topology, Geometry, and Group Theory: Informed by Experiment. October 21-25, 2013.
3. Geometric Structures in Low-Dimensional Dynamics. November 18-22, 2013.

Bill Thurston is dead at age 65.

Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years.   I don’t think it’s possible to overstate the revolutionary impact that he had on the study of geometry and topology.  Almost everything we blog about here has the imprint of his amazing mathematics.    Bill was always very generous with his ideas, and his presence in the community will be horribly missed.    Perhaps I will have something more coherent to say later, but for now here are some links to remember him by:

• Wikipedia.
• 2010 lecture on The mystery of 3-manifolds.
• On proof and progress in mathematics.

The virtual Haken conjecture

Agol’s preprint, which includes a long appendix joint with Groves and Manning, is now on the arXiv.

Dehn filling and genus dropping

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 with Heegaard genus , a torus boundary component and two slopes on such that Dehn filling along each slope produces a 3-manifold with Heegaard genus less than or equal to ?

Some thoughts on the Kirby Theorem

I don’t think it would be too controversial to assert that the Kirby Theorem is an important theorem in low dimensional topology. Given a 3-manifold and an framed link (by “framed” let’s mean “integer framed”), let denote the 3-manifold obtained from by surgery around . The Kirby Theorem states that, given two framed links , the 3-manifolds and are homeomorphic if and only if and are related by a finite sequence of the following local moves: