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Abstracts for Workshop on 4-manifolds, September 16 - 20, 2019

Alternatively have a look at the program.

On the failure of the lightbulb lemma

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Speaker: 
David Gabai
Affiliation: 
Princeton
Date: 
Mon, 2019-09-16 10:00 - 10:50
Location: 
MPIM Lecture Hall

The light bulb lemma more or less asserts, subject to a certain hypothesis, that if a surface $S$ has a transverse sphere, then a tube linking $S$ can be isotopically pulled across $S$.  Is the failure of this certain hypothesis ever interesting?  We offer some preliminary thoughts.  
 

Singular symplectic isotopy problems

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Speaker: 
Laura Starkston
Affiliation: 
UC Davis
Date: 
Mon, 2019-09-16 11:30 - 12:20
Location: 
MPIM Lecture Hall

The smooth symplectic isotopy problem asks for the classification of smooth symplectic surfaces of the complex projective plane up to isotopy. This problem remains open in degrees greater than 17. Here we will consider a singular version of the problem--classifying symplectic submanifolds of $\mathbb{CP}^2$ with singularities of specified types given by cones on torus knots. We will show that for certain rational cuspidal curves, the classification can be made completely. This is based on joint work with Marco Golla.


 

Exotic Mazur manifolds and knot trace invariants

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Speaker: 
Lisa Piccirillo
Affiliation: 
UC Austin
Date: 
Mon, 2019-09-16 15:00 - 15:50
Location: 
MPIM Lecture Hall

From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant $\nu$ is an invariant of a particular 4-manifold associated to the knot, called the knot trace. As a corollary, we produce integer homology 3-spheres admitting two distinct $S^1\times S^2$ surgeries, resolving a question from Problem 1.16 in Kirby's list.

Concordance invariants from covering involutions

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Speaker: 
András Stipsicz
Affiliation: 
Alfréd Rényi Institute, Budapest
Date: 
Tue, 2019-09-17 09:30 - 10:20
Location: 
MPIM Lecture Hall

We use the branched cover construction and ideas from connected Floer homology to define concordance invariants of knot in $S^3$. Calculations can be performed for double branched covers, in which case the invariants are trivial for alternating and torus knots and non-trivial for some pretzel knots. This allows us to derive some independence results in the smooth concordance group of knots.
 

A ribbon obstruction for Fox-colorable knots

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Speaker: 
Alexandra Kjuchukova
Affiliation: 
University of Wisconsin-Madison/MPIM
Date: 
Tue, 2019-09-17 10:30 - 10:55
Location: 
MPIM Lecture Hall

Let $K$ be a knot with the property that $\pi_1(S^3\backslash K)$ surjects onto a dihedral group. I will define a ribbon obstruction for $K$, given a cover of $S^4$ branched along a surface embedded smoothly in $S^4$ except for one cone singularity, the cone on $K$. I will give examples of knots whose non-ribbonness can be detected by this method, and I will state a few results in the subject. Based on joint works with Cahn, Geske, Orr, Shaneson.
 

Smoothly exotic spheres in four manifolds

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Speaker: 
Hannah Schwartz
Affiliation: 
Bryn Mawr/MPIM Bonn
Date: 
Tue, 2019-09-17 11:30 - 12:20
Location: 
MPIM Lecture Hall

Throughout this talk, we will compare the notions of topological isotopy, smooth isotopy, and smooth equivalence (via an ambient diffeomorphism preserving homology) between homotopic $2$-spheres smoothly embedded in a $4$-manifold. In particular, we will construct pairs of spheres that are smoothly equivalent but not even topologically isotopic. Indeed, our examples show that Gabai's recent "4D Lightbulb Theorem" does not hold without the 2-torsion hypothesis.
 

The Goeritz group and the 2-width of 3-manifolds embedded in $\mathbb{R}^4$

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Speaker: 
Michael Freedman
Affiliation: 
Station Q & UC Santa Barbara
Date: 
Tue, 2019-09-17 15:00 - 15:50
Location: 
MPIM Lecture Hall

I will discuss various notions of $k$-width for both manifolds and embeddings of manifolds in Euclidian space, with special emphasis on three dimensional manifolds and their embeddings in $\mathbb{R}^4$.  I will largely follow my recent arXiv posting on this subject.

 

Families of diffeomorphisms of 4-manifolds via graph surgery

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Speaker: 
Tadayuki Watanabe
Affiliation: 
Shimane University
Date: 
Wed, 2019-09-18 09:30 - 10:20
Location: 
MPIM Lecture Hall

In this talk, I will explain a method to construct families of diffeomorphisms of a 4-manifold by using a 4D analogue of Goussarov-Habiro's theory of graph surgery in 3D. Our graph surgery would produce lots of potentially nontrivial elements of homotopy groups of diffeomorphism groups of 4-manifolds. I will discuss about their applications to 4D analogue of the Smale conjecture and the 4D light bulb theorem for 3-disk.

 

Intersection forms of definite manifolds bounded by lens spaces

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Speaker: 
JungHwan Park
Affiliation: 
Georgia Institute of Technology (GIT)
Date: 
Wed, 2019-09-18 10:30 - 10:55
Location: 
MPIM Lecture Hall

We find some restrictions on the intersection forms of smooth definite manifolds bounded by rational homology spheres which are rationally cobordant to lens spaces. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite rank cokernels. Further, we show that there is no $n$ such that every lens space smoothly embeds in $n$ copies of the complex projective plane. This is joint work with Paolo Aceto and Daniele Celoria.
 

Hidden Algebraic Structure in Topology

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Speaker: 
Sergei Gukov
Affiliation: 
Caltech
Date: 
Wed, 2019-09-18 11:30 - 12:20
Location: 
MPIM Lecture Hall

Which 4d TQFTs and  4-manifold invariants detect the Gluck twist? Guided by questions like this, we will look for new invariants of smooth 4-manifolds and knotted surfaces in 4-manifolds.

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