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Abstracts for Conference on 4-manifolds and knot concordance

Alternatively have a look at the program.

4-dimensional $L^2$-acyclic bordism and Whitney towers

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Speaker: 
Jae Choon Cha
Affiliation: 
POSTECH
Date: 
Mon, 2016-10-17 09:30 - 10:30
Location: 
MPIM Lecture Hall

Recently Sylvain Cappell, Jim Davis and Shmuel Weinberger studied $L^2$-acyclic bordism groups in high dimensions.  In this talk we address 4-dimensional $L^2$-acyclic bordism between 3-manifolds.  We introduce a Whitney tower approach to study the structure in this dimension, and show that Cheeger-Gromov invariants over amenable groups give obstructions.  Also we answer some questions of Cappell, Davis and Weinberger which concern the relationship of $L^2$-acyclic bordism and knot concordance.

Whitney towers: What are they? And what are they good for?

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Speaker: 
Rob Schneiderman
Affiliation: 
Lehman College, City University of New York
Date: 
Mon, 2016-10-17 11:00 - 12:00
Location: 
MPIM Lecture Hall

This talk will sketch an introduction to current theories of Whitney towers, highlight recent results and applications, and point out some open problems.

Knot concordance in homology spheres

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Speaker: 
Jennifer Hom
Affiliation: 
Georgia Tech
Date: 
Mon, 2016-10-17 13:30 - 14:30
Location: 
MPIM Lecture Hall

The knot concordance group $\mathcal{C}$ consists of knots in $S^3$ modulo knots that bound smooth disks in $B^4$. We consider $\widehat{\mathcal{C}}_{\mathbb{Z}}$, the group of knots in homology spheres that bound homology balls modulo knots that bound smooth disks in a homology ball. Matsumoto asked if the natural map from $\mathcal{C}$ to $\widehat{\mathcal{C}}_{\mathbb{Z}}$ is an isomorphism. Adam Levine answered this question in the negative by showing the map is not surjective.

Embedding of 3-manifolds in spin 4-manifolds

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Speaker: 
Kyle Larson
Affiliation: 
Michigan State University
Date: 
Mon, 2016-10-17 14:45 - 15:15
Location: 
MPIM Lecture Hall

An invariant of orientable 3-manifolds is defined by taking the minimum $n$ such that a given 3-manifold embeds in the connected sum of $n$ copies of $S^2\times S^2$, and we call this $n$ the \emph{embedding number} of the 3-manifold. We discuss some general properties of this invariant, and discuss some calculations for families of lens spaces and Brieskorn spheres. We can construct rational and integral homology spheres whose embedding numbers grow arbitrarily large, and which can be calculated by exactly if we assume the $11/8$-Conjecture.

The slice- ribbon conjecture and related topics.

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Speaker: 
Tetsuya Abe
Affiliation: 
Osaka City University Advanced Mathematical Institute
Date: 
Mon, 2016-10-17 15:30 - 16:00
Location: 
MPIM Lecture Hall

We survey the recent progress on the slice-ribbon conjecture, which is one of the biggest conjectures in knot concordance theory. In this talk, we will give some potential counterexamples of the slice-ribbon conjecture using annulus twist, which is a certain operation on knots. Also, we will discuss related topics.
 

The mod 8 signature of a surface bundle

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Speaker: 
Andrew Ranicki
Affiliation: 
University of Edinburgh
Date: 
Mon, 2016-10-17 16:30 - 17:00
Location: 
MPIM Lecture Hall

The talk will report on the current status of a joint project with Dave Benson, Caterina Campagnolo and Carmen Rovi. Werner Meyer (Bonn thesis 1972) proved that the signature $\sigma(E) \in \mathbb{Z}$ of a surface bundle $\Sigma_g \to E \to \Sigma_h$ is divisible by 4, and can be computed from a  cohomology class $\tau \in H^2(\operatorname{Sp}(2g,\mathbb{Z});\mathbb{Z})$ and the monodromy $\pi_1(\Sigma_h) \to \operatorname{Sp}(2g,\mathbb{Z})$.

Searching for slice alternating knots

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Speaker: 
Brendan Owens
Affiliation: 
University of Glasgow
Date: 
Tue, 2016-10-18 09:30 - 10:30
Location: 
MPIM Lecture Hall

Which alternating knots are slice?  Which are ribbon?  For which does the double branched cover bound a rational homology ball?  Unfortunately I can't answer any of these questions.  I will report on a computer search with Frank Swenton which found approximately 30,000 new examples of slice alternating knots, and discuss some related questions.

Functions on surfaces and constructions of 3-, 4- and 5-manifolds

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Speaker: 
David Gay
Affiliation: 
University of Georgia
Date: 
Tue, 2016-10-18 11:00 - 12:00
Location: 
MPIM Lecture Hall

I'll steal ideas from Lickorish's proof that 3-manifolds bound 4-manifolds and Hatcher and Thurston's proof that the mapping class group of a surface is finitely presented to give, among other things, a new proof that $\Omega_4=\mathbb{Z}$ (using trisections where Lickorish uses Heegaard splittings). The key idea is that generic $n$-parameter families of functions on surfaces describe  $(n+3)$-manifolds, at least for $n\leq 2$.

Group actions on exotic smoothings of $\mathbb{R}^4$

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Speaker: 
Robert Gompf
Affiliation: 
The University of Texas at Austin
Date: 
Tue, 2016-10-18 13:30 - 14:30
Location: 
MPIM Lecture Hall

It has been known for more than three decades that there are uncountably many diffeomorphism types of smoothings on $\mathbb{R}^4$, so in some sense there is a shortage of diffeomorphisms. However, nothing has been known about the self-diffeomorphisms of such a manifold up to smooth isotopy, except that there is only one (preserving orientation) for the standard smoothing. This talk will cover a recent breakthrough in the matter.

Lower order quotients in the $n$-solvable filtration

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Speaker: 
Taylor Martin
Affiliation: 
Sam Houston State University
Date: 
Tue, 2016-10-18 14:45 - 15:15
Location: 
MPIM Lecture Hall

We establish several results about two short exact sequences involving lower terms of the $n$-solvable filtration, $\{\mathcal{F}_n^m\}$ of the string link concordance group $\mathcal{C}^m$, which was introduced by Cochran, Orr, and Teichner in the late 90's. We show that the short exact sequence
\[0\to \mathcal{F}_0^m/\mathcal{F}_{0.5}^m\to \mathcal{F}_{-0.5}^m/\mathcal{F}_{0.5}^m\to\mathcal{F}_{-0.5}^m/\mathcal{F}_{0}^m\to  0\]

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