# Columbia Geometric Topology Seminar

## Fall 2010

The GT seminar meets on Fridays in Math. 520, at 1:20PM.
Organizer: Walter Neumann.
Other area seminars. Our e-mail list. Archive of previous semesters

### FALL 2010

Date Speaker Title
September 10 Everyone Organizational meeting joint with SGGTC seminar
September 17 Walter Neumann What does a complex surface really look like?
September 24 Peter Storm Infinitesimal rigidity of hyperbolic manifolds with totally geodesic boundary
October 1 Daniel Mathews Hyperbolic cone-manifolds with prescribed holonomy
October 8 Adam Sikora  Quantization of Character Varieties
October 15  Alexander Gaifullin Local combinatorial formulae for Pontryagin classes
October 22  Jozef Przytycki  Homology of distributive structures
October 29  Eriko Hironaka  The monodromies of a fibered 3-manifold
November 5  Kevin Walker Higher dimensional Deligne conjecture and the blob complex
Wed Nov 10 1:00pm Room 622
NOTE TIME, DATE, ROOM
Julien Marché
(also speaking Nov 9 at CUNY Grad Center)
A geometric interpretation of combinatorial formulas for Chern-Simons invariants
November 12  Tetsuya Ito An application of orderings of Mapping class groups
November 19  Johanna Mangahas Geometry of right-angled Artin groups in mapping class groups
November 26  No Seminar  University Holiday
December 3, 10:45am
Room 417
Joan Birman Pseudo-Anosov mapping classes with minimum dilatation
December 10, Double header 1:15pm  Brendan Owens Alternating links and rational balls
December 10, Double header 2:30pm  Roland van der Veen Hyperbolic polyhedra and the Jones polynomial
Tuesday December 21 4:30pm  Martin Bridson  Actions of higher-rank lattices on surfaces and free groups

### Spring 2011

Date Speaker Title
January 14 Dani Wise Morse Theory, Random Subgraphs, and Incoherent Groups
January 21 Everyone Organizational meeting
January 28 Liam Watson Left-orderability and Dehn surgery
February 4 Josh Greene Conway mutation of alternating links
February 11 No seminar
February 18 John Etnyre The contact sphere theorem and tightness in contact metric manifolds
February 25, 1:00 PM David Shea Vela-Vick Pontryagin invariants and integral formulas for Milnor's triple linking number
March 4 Lenhard Ng Filtered knot contact homology and transverse knots
March 11 No seminar
March 18 Spring break; no seminar
March 25 Sucharit Sarkar Grid diagrams and the Ozsvath-Szabo tau invariants
April 1 Dylan Thurston A faithful action of the mapping class group from Heegaard Floer homology
April 8 Martin Scharlemann Fibered knots and potential counterexamples to the Property 2R and Slice-Ribbon Conjectures
April 15 Nathan Dunfield The least spanning area of a knot and the Optimal Bounding Chain Problem
April 22 Jacob Rasmussen Khovanov homology of torus knots
April 29 Ken Baker Bridge numbers and Dehn surgery
May 6 (First day of finals)

## Abstracts.

#### Sept 17, 2010.

Walter Neumann, “What does a complex surface really look like?”

A complex surface embedded in some C^n inherits a geometry which can be surprisingly complicated locally. I will describe this local geometry in terms of the JSJ decomposition of a neighborhood boundary of a point. This is joint work with Anne Pichon and Lev Birbrair (with input from Don O'Shea and others), and is ongoing work in the sense that we know what the picture is, but not all details are proved yet.

#### Sept 24, 2010.

Peter Storm, Jane Street, “Infinitesimal rigidity of hyperbolic manifolds with totally geodesic boundary”

Using the Bochner technique, Steve Kerckhoff and I recently proved the following theorem. Let M be a compact hyperbolic manifold with totally geodesic boundary. If M has dimension at least four, then the holonomy representation of M is infinitesimally rigid. This is an infinite volume analog of the Calabi-Weil rigidity theorem. I will explain some of the background and ideas used in the proof.

#### October 1, 2010.

Daniel Mathews, Boston College “Hyperbolic cone-manifolds with prescribed holonomy”

We examine the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A geometric cone-manifold structure on a surface, with all interior cone angles being integer multiples of 2π, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a geometric cone-manifold structure. Our constructions involve the Euler class of a representation, the universal covering group of the orientation-preserving isometries of the hyperbolic plane, and the action of the outer automorphism group on the character variety.

#### October 8, 2010.

Adam Sikora, SUNY Buffalo (visiting Columbia this semester), “Quantization of Character Varieties”

We prove that G-character varieties of surfaces are completely integrable systems for Lie groups of rank <=2. We explain the motivation for this result coming from the program of defining Reshetikhin-Turaev-Witten quantum invariants via the process of geometric quantization of character varieties.

#### October 15, 2010.

Alexander Gaifullin, Moscow State University and Steklow Institute, “Local combinatorial formulae for Pontryagin classes”

The talk will be devoted to the problem of combinatorial computation of the rational Pontryagin classes of a triangulated manifold. This problem goes back to the famous work by A.M. Gabrielov, I.M. Gelfand, and M.V. Losik (1975). Since then several different approaches to combinatorial computation of the Pontryagin classes have been suggested. However, these approaches require a combinatorial manifold to be endowed with some additional structure such as smoothing or certain its discrete analogue. We suggest a new approach based on the concept of a universal local formula. This approach allows us to construct an explicit combinatorial formula for the first Pontryagin class that can be applied to any combinatorial manifold without any additional structure.

#### October 22, 2010.

Jozef Przytycki, GWU, “Homology of distributive structures”

Homology theory of associative structures like groups and rings has been zealously studied throughout the past starting from the work of Hopf, Eilenberg, and Hochschild, but non-associative structures, like quandles, were neglected till recently.
The condition which can often replace associativity is distributivity and my talk will be devoted to homology of distributive structures with an eye on a hypothetical connection to Khovanov homology.

#### October 29, 2010.

Eriko Hironaka, FSU, “The monodromies of a fibered 3-manifold”

In this talk we fix a hyperbolic 3-manifold M and consider all its fibrations over the circle. Thurston described a way to parameterize the fibrations as integer lattice points in a union of fibered cones". We will describe properties of mapping classes within a fibered cone, including the shapes of invariant train tracks and the corresponding shape" of the defining polynomials of the dilatation. As an application, we find mapping classes whose normalized dilatations converge to 1 + golden mean.

#### November 5

Kevin Walker, “Higher dimensional Deligne conjecture and the blob complex”

The blob construction associates a chain complex to a pair (M, C), where M is an n-manifold and C is an n-category. From a combinatorial topology point of view, this can be thought of as derived category generalization of a skein module (equivalently, of a TQFT Hilbert space). From a homological algebra point of view, it can be thought of as a more geometric version of the Hochschild complex which also works for n-categories. (The usual Hochschild complex is the case n=1 and M = S^1.)
Deligne's conjecture gives an action of chains of the little 2-disks operad on Hochschild cochains. I will describe an n-dimensional generalization of this action, where Hochschild cochains are replaced by maps between blob complexes and the little 2-disks operad is replaced by what one might call the little n-surgery operad, which contains the little (n+1)-ball operad. Even in the case n=1 it goes beyond the usual Deligne conjecture and gives an action of an operad composed of higher genus surfaces.
This is joint work with Scott Morrison.

#### November 10 1:00pm

Julien Marché, “A geometric interpretation of combinatorial formulas for Chern-Simons invariants”

The Chern-Simons invariant of 3-manifolds equipped with a representation in PSL(2,C) is a secondary characteristic class generalizing the volume of a hyperbolic 3-manifold. J. Dupont and W. Neumann explained how this invariant can be computed from an ideal triangulation in terms of the Thurston coordinates. I will explain a derivation of their formula which does not involve the (extended) Bloch group but is more directly linked with the initial definition of Chern and Simons.

#### November 12

Tetsuya Ito, “An application of orderings of Mapping class groups”

The mapping class group of surfaces with non-empty boundary has left-invariant orderings, called Thurston-type orderings. In this talk, I will explain how the orderings are related to the topology and geometry of 3-manifolds via the open book decomposition. I will show the relationships between orderings and contact structures, or surfaces in openbooks.

#### November 19

Johanna Mangahas “Geometry of right-angled Artin groups in mapping class groups”

I'll describe joint work with Matt Clay and Chris Leininger. We give sufficient conditions for a finite set of mapping classes to generate a right-angled Artin group. This subgroup is quasi-isometrically embedded in the whole mapping class group, as well as, via the orbit map, in Teichmueller space with either of the standard metrics. Subsurface projection features prominently in the proofs.

#### December 3, 10:45am

Joan Birman“Pseudo-Anosov mapping classes with minimum dilatation”

Let $S$ be a closed orientable surface of genus $g \geq 7$. Consider the set of all real numbers $\lambda_f > 1$ such that $\lambda_f$ is the dilatation of a pseudo-Anosov map $f:S \to S$. It is well known that this set has a greatest lower bound $\lambda_{min,S}$, and that $\lambda_{min,S}$ is achieved for some pseudo-Anosov map $f:S\to S$. The main result in this paper is to prove that for each genus $g$ there is a finite, enumerable set of possibilities for $\lambda_{min,S}$.

#### December 10

Brendan Owens “Alternating links and rational balls”

For a slice knot K in the 3-sphere it is well known that the double branched cover Y_K bounds a smooth rational homology 4-ball. Paolo Lisca has shown that this condition is sufficient to determine sliceness for 2-bridge knots, and that this generalises to 2-bridge links. I will discuss the problem of determining whether Y_L bounds a rational ball when L is an alternating link.

#### December 10

Roland van der Veen “Hyperbolic polyhedra and the Jones polynomial”

For knots the hyperbolic geometry of the complement is known to be related to its Jones polynomial in various ways. We propose to study this relationship more closely by extending the Jones polynomial to graphs. For a planar graph we will show how its Jones polynomial then gives rise to the hyperbolic volume of the polyhedron whose 1-skeleton is the graph. Joint with Francois Gueritaud and Francois Costantino

#### Tuesday December 21, 4:30pm

Martin Bridson “Actions of higher-rank lattices on surfaces and free groups”

Following a brief discussion of rigidity for mapping class groups and automorphism groups of free groups, I shall outline the proof of a recent result with Ric Wade: every homomorphism from a higher-rank lattice to the outer automorphism group of a free group has finite image.

#### January 14

Dani Wise “Morse Theory, Random Subgraphs, and Incoherent Groups”

I will use the "probabilistic method" to prove that certain polygons of finite groups have finitely generated subgroups that are not finitely presented. This gives a new family of incoherent Kleinian groups. I'll try to give an idea of what I'm doing here, and perhaps what to expect next.

#### January 28

Liam Watson “Left-orderability and Dehn surgery”

Left-orderability of groups is a property that seems to have a geometric flavour. For example, in the context 3-manifolds, this property on the fundamental group is related to taut foliations. It is known that the fundamental group of a knot complement is always left-orderable, however the result of Dehn surgery along the knot yields a group that need not inherit this property. This talk investigates the relationship between left-orderability and Dehn surgery, and exhibit knots having the property that all sufficiently positive surgeries yield manifolds with non-left-orderable fundamental groups. These knots turn out to all be L-space knots, and in light of the fact that all known examples of L-spaces have non-left-orderable fundamental group, it is interesting to compare this result with the fact that L-space surgeries behave in an analogous manner. This is joint work with Adam Clay.

#### February 4

Josh Greene “Conway mutation of alternating links”

Let D and D' denote connected, reduced, alternating diagrams for a pair of links, and Y and Y' their branched double-covers. I'll discuss the proof and consequences of the following result: Y and Y' have isomorphic Heegaard Floer homology groups iff Y and Y' are homeomorphic iff D and D' are mutants.

#### February 18

John Etnyre “The contact sphere theorem and tightness in contact metric manifolds”

We establish an analog of the sphere theorem in the setting of contact geometry. Specifically, if a given three dimensional contact manifold admits a compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure is tight. The proof is a blend of topological and geometric techniques. A necessary technical result is a lower bound for the radius of a tight ball in a contact 3-manifold. We will also discuss geometric conditions in dimension three for a contact structure to be universally tight in the nonpositive curvature setting. This is joint work with Rafal Komendarczyk and Patrick Massot.

#### February 25

David Shea Vela-Vick “Pontryagin invariants and integral formulas for Milnor's triple linking number”

To each three-component link in the 3-dimensional sphere we associate a characteristic map from the 3-torus to the 2-sphere, and establish a correspondence between the pairwise and Milnor triple linking numbers of the link and the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere.

#### March 4

Lenhard Ng “Filtered knot contact homology and transverse knots”

Knot contact homology is a combinatorial Floer-theoretic knot invariant derived from Symplectic Field Theory. I'll discuss a new filtered version of this theory, transverse homology, which is an invariant of transverse knots, and I'll try to compare its effectiveness to the effectiveness of the transverse invariant in knot Floer homology.

#### April 8

Martin Scharlemann “Fibered knots and potential counterexamples to the Property 2R and Slice-Ribbon Conjectures”

If there are any two component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions. The simplest plausible counterexample to the Generalized Property R Conjecture could be a two component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to (S1 × S2) # (S1 × S2). We exhibit a family of such links that are probably counterexamples to Generalized Property R. These links can be used to generate slice knots that do not seem to be ribbon.

#### April 15

Nathan Dunfield “The least spanning area of a knot and the Optimal Bounding Chain Problem”

Two fundamental objects in knot theory are the minimal genus surface and the least area surface bounded by a knot in a 3-dimensional manifold. When the knot is embedded in a general 3-manifold, the problems of finding these surfaces were shown to be NP-complete and NP-hard respectively. However, there is evidence that the special case when the ambient manifold is R^3, or more generally when the second homology is trivial, should be considerably more tractable. Indeed, we show here that a natural discrete version of the least area surface can be found in polynomial time. The precise setting is that the knot is a 1-dimensional subcomplex of a triangulation of the ambient 3-manifold. The main tool we use is a linear programming formulation of the Optimal Bounding Chain Problem (OBCP), where one is required to find the smallest norm chain with a given boundary. While the decision variant of OBCP is NP-complete in general, we give conditions under which it can be solved in polynomial time. We then show that the least area surface can be constructed from the optimal bounding chain using a standard desingularization argument from 3-dimensional topology. We also prove that the related Optimal Homologous Chain Problem is NP-complete for homology with integer coefficients, complementing the corresponding result of Chen and Freedman for mod 2 homology.

This is joint with Anil Hirani.

#### April 22

Jacob Rasmussen “Khovanov homology of torus knots”

I'll describe some conjectures about the structure of Khovanov homology of torus knots. They are motivated by work of Oblomkov and Shende in algebraic geometry, but can be stated purely in terms of representation theory. This is joint work with E. Gorsky, A. Oblomkov, and V. Shende.

#### April 29

Ken Baker “Bridge numbers and Dehn surgery”

Dehn surgery on a knot K' in S3 produces a new knot K in a manifold M. What can be said about the bridge number of K with respect to a Heegaard splitting of M? We'll first overview known results and results-in-progress for when the surgery slope has large distance from the meridian of K'. Then we'll focus on a contrasting family of examples of distance 1 surgeries. This is joint work with Cameron Gordon and John Luecke.

# Other relevant information.

## Previous semesters:

Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.

## Our e-mail list.

Announcements for this seminar, as well as for related seminars and events, are sent to the GT seminar mailing list. You can subscribe directly or by contacting Walter Neumann.