# Columbia Geometric Topology Seminar

Fall 2019

Organizer: Nick Salter.
The GT seminar meets on Fridays at 2:00pm Fridays in room 520. We also have an overflow room 622 from 11 to 1 Fridays for additional talks.

Other area seminars. Our e-mail list. Archive of previous semesters

## Fall 2019

Date Speaker Title
Sept 13 Indira Chatterji Group actions on median spaces and generalizations
Sept 20 Ty Ghaswala Promoting circle actions to actions on the real line
Sept 27 Danny Calegari Big mapping class groups and rigidity of the simple circle
Oct 4 Khalid Bou-Rabee Quantifying residual finiteness
Oct 11 Dusa McDuff Counting curves with tangency constraints in the complex projective plane
Oct 18 Margaret Nichols Taut sutured handlebodies as twisted homology products
Oct 25 Francesco Lin Hyperbolic four-manifolds with vanishing Seiberg-Witten invariants
Nov 1 Abdul Zalloum CAT(0) groups are determined by sublinear boundaries
Nov 8 Nick Vlamis Topology of (big) mapping class groups
Nov 15 Kyle Hayden Knots, spines, and exotic 4-manifolds
Nov 22 Diana Hubbard Right-veering open books and the Upsilon invariant
Dec 6 Jingyin Huang The Helly geometry of some Garside and Artin groups
Dec 13 Maggie Miller Light bulbs in 4-manifolds

## Abstracts

Maggie Miller, Princeton

NOTE: TALK IS IN ROOM 622

December 13, 2019
Title: Light bulbs in 4-manifolds

Abstract:
In 2017, Gabai proved the light bulb theorem, showing that if $R$ and $R'$ are 2-spheres homotopically embedded in a 4-manifold with a common dual, then with some condition on 2-torsion in $\pi_1(X)$ one can conclude that $R$ and $R'$ are smoothly isotopic. Schwartz gave examples showing that this 2-torsion condition is necessary, and Schneiderman and Teichner then obstructed the isotopy whenever this condition fails, yielding a complete solution to when light bulbs are isotopic. I weakened the hypothesis on $R$ and $R'$ by allowing $R'$ not to have a dual and showed that the spheres are still smoothly concordant.

I will talk about these various definitions and theorems as well as current joint work with Michael Klug obstructing concordance in the case that the condition on 2-torsion in $\pi_1(X)$ fails, yielding a complete solution to when $R$ and $R'$ are concordant in this generalized setting.

Part of this work is joint with Michael Klug.

References: https://arxiv.org/abs/1903.03055

Jingyin Huang, Ohio State

December 6, 2019
Title
The Helly geometry of some Garside and Artin groups
Abstract:
Garside groups and Artin groups are two generalizations of braid groups. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, hence equip these groups with a particular nonpositive-curvature-like (NPC) structure. Such structure shares many properties of a CAT(0) structure and has some additional combinatorial flavor. We shall explain this NPC structure in more detail and discuss new results on the topology and geometry of these groups which are immediate consequences of such structure. This is joint work with D. Osajda.

References: https://arxiv.org/abs/1904.09060

Diana Hubbard, CUNY

November 22, 2019
Title
Right-veering open books and the Upsilon invariant
Abstract:
Fibered knots in a three-manifold Y can be thought of as the binding of an open book decomposition for Y. A basic question to ask is how properties of the open book decomposition relate to properties of the corresponding knot. In this talk I will describe joint work with Dongtai He and Linh Truong that explores this: specifically, we can give a sufficient condition for the monodromy of an open book decomposition of a fibered knot to be right-veering from the concordance invariant Upsilon.  I will discuss some applications of this work, including an application to the Slice-Ribbon conjecture.

Kyle Hayden, Columbia

November 15, 2019
Title
Knots, spines, and exotic 4-manifolds
Abstract:
A "knot trace" is a simple 4-manifold obtained by attaching a thickened disk (i.e. a 2-handle) to the 4-ball along a knot in its boundary. These 4-manifolds arise in many contexts, including the construction of exotic smooth structures on 4-space. After a quick survey of this history, I'll use knot traces to present a simple alternative solution to a classical question about piecewise linear surfaces in 4-manifolds (Problem 4.25 on Kirby's list, recently resolved by Levine and Lidman). The underlying construction provides pairs of homeomorphic 4-manifolds X and X' such that X is a knot trace and yet X' is not diffeomorphic to any knot trace, including X. These "not knot traces" X' exhibit a variety of other curious phenomena; I'll briefly mention some applications to the study of handle decompositions of simply-connected 4-manifolds, Stein fillings of contact 3-manifolds, and knot concordance. This is joint work in progress with Lisa Piccirillo.

Nick Vlamis, CUNY

November 8, 2019
Title: Topology of (big) mapping class groups

Abstract:
Mapping class groups inherit a natural topology from the compact-open topology on homeomorphism groups.  When the underlying surface is of infinite type, this topology is no longer discrete, which allows us to study these mapping class groups from the perspective of topological group theory.  In the talk, I will discuss two results, both of which highlight applications of this topological group theoretic perspective.

References: https://arxiv.org/abs/1711.03132

Abdul Zalloum, Queens University

November 1, 2019
Title: CAT(0) groups are determined by sublinear boundaries

Abstract:
To each hyperbolic space, one can associate a space at infinity called the Gromov’s boundary. Gromov showed that a quasi-isometry between two hyperbolic spaces induces a homeomorphism on their boundaries. For a CAT(0) space, one can also assign a space at infinity called the visual boundary but that is no longer a quasi-isometry invariant. Several attempts have been made to circumvent the problem, most recent of which is work by Qing and Rafi. They introduce the notion of a ”sublinear contracting boundary” of a CAT(0) space and they show that a quasi-isometry between two CAT(0) spaces induces a homeomorphism between their sublinear boundaries. We investigate when the converse holds: Given a homeomorphism between two sublinear boundaries of CAT(0) spaces, when is it induced by a quasi isometry? We show that a homeomorphism f between two cocompact CAT(0) spaces f:X-->Y is induced by a quasi-isometry if and only if f is stable and Morse quasi-Mobius.  In this talk, I will define all the objects above and give a sketch for the proof. This is joint work with Yulan Qing.

Francesco Lin, Columbia

October 25, 2019
Title
Hyperbolic four-manifolds with vanishing Seiberg-Witten invariants
Abstract:
We show the existence of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants, addressing a conjecture of Claude LeBrun. This is achieved by showing, using results in geometric and arithmetic group theory, that certain hyperbolic 4-manifolds contain L-spaces as hypersurfaces. This is joint work with Ian Agol.

Reference:https://arxiv.org/abs/1812.06536

Margaret Nichols, SUNY Buffalo

October 18, 2019
Title
Taut sutured handlebodies as twisted homology products
Abstract:
A basic problem in the study of 3-manifolds is to determine when geometric objects are of ‘minimal complexity’. We are interested in this question in the setting of sutured manifolds, where minimal complexity is called ‘tautness’.

One method for certifying that a sutured manifold is taut is to show that it is homologically simple - a so-called ‘rational homology product’. Most sutured manifolds do not have this form, but do always take the more general form of a ‘twisted homology product’, which incorporates a representation of the fundamental group. The question then becomes, how complicated of a representation is needed to realize a given sutured manifold as such?

We explore the case of sutured handlebodies, and see even among the simplest class of these, twisting is required. We give examples that, when restricted to solvable representations, the twisting representation cannot be ‘too simple’.

Dusa McDuff, Columbia

October 11, 2019
Title
Counting curves with tangency constraints in the complex projective plane
Abstract:
The attempt to prove the stabilized ellipsoidal symplectic embedding problem has lead to some very interesting questions about the behavior of genus zero pseudo-holomorphic curves in $\C P^2$.  In particular, it seems that one can generalize the Caporaso--Harris recursion formula that counts curves tangent to a global divisor to cases where one also allows tangencies to local divisors.  I will explain the embedding problem, the relevance of curve counts to it, and then explain some recent joint work with Kyler Siegel that develops new ways to perform these counts.   The talk will assume no special knowledge of symplectic geometry.

Khalid Bou-Rabee, CUNY

October 4, 2019
Title
Quantifying residual finiteness
Abstract:
The theory of quantifying residual finiteness assigns, to each finitely generated group, an invariant that indicates how well-approximated the group is by its finite quotients. We introduce this theory and survey the current state of the subject. There will be a strong emphasis on examples, open questions, and connections to other subjects.

Danny Calegari, University of Chicago

September 27, 2019
Title: Big mapping class groups and rigidity of the simple circle
Abstract:
Let G denote the mapping class group of the plane minus a Cantor set. We show that every action of G on the circle  is either trivial or semi-conjugate to a unique minimal action on the so-called simple circle. This is joint work with Lvzhou (Joe) Chen.

Ty Ghaswala, University of Manitoba

September 20, 2019
TitlePromoting circle actions to actions on the real line
Abstract
Circularly-orderable and left-orderable groups play an important, and sometimes surprising, role in low-dimensional topology and geometry. For example, these combinatorial conditions completely characterize when a countable group acts on a 1-manifold. Through the so-called L-space conjecture, left-orderability of the fundamental group of a rational homology 3-sphere is connected to the existence of taut foliations and properties of its Heegaard Floer homology. I will present new necessary and sufficient conditions for a circularly-orderable group to be left-orderable, and introduce the obstruction spectrum of a circularly-orderable group. This raises a plethora of intriguing questions, especially in the case when the group is the fundamental group of a manifold.

This is joint work with Jason Bell and Adam Clay.

Indira Chatterji, CNRS
September 13, 2019
TitleGroup actions on median spaces and generalizations
Abstract
Median spaces are a natural generalization of R-trees and CAT(0) cubical complexes. I will define the notion, discuss the context and show that the fundamental group of a compact hyperbolic manifold acts properly and cocompactly on a median space.