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The 11th William Rowan Hamilton Geometry and Topology Workshop

on

Interactions between three and four manifolds

August 25-29, 2015

The Hamilton Mathematics Institute, Trinity College Dublin


Mini-Course

on

Bridging classical three and four manifold theory

with

Martin Scharlemann (Santa Barbara)

and

David Gay (Georgia)

August 25-26, 2015

Schedule of Mini-Course:

All minicourse lectures will be held in the Joly Lecture Hall in the Hamilton Building (see campus map).


Time
Speaker
Talk
Tuesday

9:30 - 10:30
David Gay (Georgia) Decompositions of low-dimensional manifolds: Handles, Heegaard splittings, Trisections
10:30 - 11:00
COFFEE & DISCUSSION
11:00 - 12:00
David Gay (Georgia) Morse functions, Cerf theory and Morse 2-functions
12:00 - 2:30
LUNCH
2:30 - 3:30
Martin Scharlemann (Santa Barbara) Codimension one embeddings: history and handle theory
3:30 - 4:00
DISCUSSION
4:00-5:00
Martin Scharlemann (Santa Barbara) Reimbedding strategies for the Schoenflies Conjecture
Wednesday

9:30 - 10:30 Martin Scharlemann (Santa Barbara) Heegaard unions, Property R, and the Akbulut-Gompf example
10.30 - 11:00
COFFEE & DISCUSSION
11:00 - 12:00 Martin Scharlemann (Santa Barbara) Meditation on generalized Property R
12:00 - 2:30 LUNCH
2:30 - 3:30 David Gay (Georgia) Trisection questions inspired by Heegaard splitting theorems
3:30 - 4:00 DISCUSSION
4:00 - 5:00
David Gay (Georgia) Open books and Lefschetz fibrations



Lecture Series on

Interactions between three and four manifolds

August 27-29, 2015

Schedule of Talks:

All talks will be held in the Joly Lecture Hall in the Hamilton Building, TCD (see campus map).

Time
Speaker
Talk
Thursday
9.00-9.45
Abigail Thompson (UC Davis) Incompressible surfaces and distance 2 manifolds
9.45-10.15
COFFEE
10:15-11:00
Paolo Lisca (Pisa) On 3-braid knots of finite concordance order
11:15-12:00 Duncan McCoy (Glasgow) Alternating knots with unknotting number one
12:00-2:30
LUNCH
2:30-3:15
Jen Hom (Columbia) Surgery obstructions from Heegaard Floer homology
3:30-4:15
Saso Strle (Ljubljana) Essential non-orientable surfaces in homology cobordisms and closed 4-manifolds
4:30-5:15
Dave Gabai (Princeton)
On slice missing slices
Friday
9:00-9:45 Peter Feller (Boston College) Smooth slice genus vs. topological slice genus
9.45-10.15 COFFEE
10:15-11:00 Frank Quinn (Virginia Tech) Some decompositions of smooth 4-manifolds
11:15-12:00 Joan Licata (Australian N.U.) Morse structures on open books
12:00-2:30 LUNCH
2:30-3:15 Rob Kirby (Berkeley) Trisections of 4-manifolds
3:30-4:15 Alex Zupan (Nebraska-Lincoln) Trisections of 4-manifolds and Dehn surgery
Saturday
9:00-9:45 Stefan Friedl (Regensburg) Fox calculus and the Thurston norm
9.45-10.15 COFFEE
10:15-11:00 Jennifer Schultens (UC Davis) Kakimizu complexes
11:15-12:00 Slava Krushkal (Virginia) Engel relations in 4-manifold topology
END OF WORKSHOP







Abstracts for Lecture Series


Peter Feller (Boston College)
Title:Smooth slice genus vs. topological slice genus.
Abstract:Smooth and topological four-manifolds are well-known to be quite different.A knot theoretic incarnation of this is the difference between the smooth and the topological slice genus. We examine this difference from two points of view. In the first part of the talk we discuss general bounds for the slice genera in terms of other knot invariants; including the following result:the degree of the Alexander polynomial is an upper bound for the topological slice genus.In the second part of the talk we focus on two particularly simple classes of knots: torus knots and two-bridge knots.

Stefan Friedl (Regensberg)
Title: Fox calculus and the Thurston norm
Abstract: If N is a 3-manifold such that its fundamental group has a presentation with two generators and one relator we will show how to obtain the Thurston norm of N from Fox calculus.

Jen Hom (Columbia)
Title: Surgery obstructions and Heegaard Floer homology
Abstract: Lickorish and Wallace proved that every closed oriented three-manifold can be expressed as surgery on a link in the three-sphere. This naturally leads to the question of which three-manifolds are surgery on a knot. We give an obstruction coming from Heegaard Floer homology, and use it to construct infinitely many small Seifert fibered integer homology spheres which are not surgery on a knot. This is joint work with Cagri Karakurt and Tye Lidman.

Robion Kirby (Berkeley)
Title: Trisections of 4-manifolds
Abstract:  Topics that are independent of David Gay's lectures in his minicourse.

Slava Krushkal (Virginia)
Title: Engel relations in 4-manifold topology
Abstract: The 2-Engel relation has been classically studied in finite and Lie groups, dating back to the work of Burnside. I will describe new applications in link theory and 4-manifold topology, focusing on the 4-dimensional topological surgery conjecture. (Joint work with Michael Freedman)

Joan Licata (Australian NU)
Title: Morse Structures on Open Books
Abstract: Every contact 3-manifold is locally contactomorphic to the standard contact R^3, but this fact does not necessarily produce large charts that cover the manifold efficiently. I'll describe joint work with Dave Gay which uses an open  book decomposition of a contact manifold to produce a particularly efficient collection of such contactomorphisms, together with simple combinatorial data describing how  to reconstruct the contact 3-manifold from these charts.   We use this construction to define  front projections for Legendrian knots and links in arbitrary contact 3-manifolds, generalising existing constructions of front projections for Legendrian knots in S^3 and universally tight lens spaces. 

Paolo Lisca (Pisa)
Title: On 3-braid knots of finite concordance order.
Abstract: The slice-ribbon conjecture was established for 2-bridge knots and infinite families of pretzel and Montesinos knots. The methods leading to these results were also used to determine the smooth concordance order of 2-bridge knots and infinitely many pretzel knots. After recalling basic definitions I will describe the methods mentioned above, and then sketch how to use them to prove that a knot of finite smooth concordance order which is the closure of a 3-braid is either ribbon or alternating.

Duncan McCoy (Glasgow)
Title: Alternating knots with unknotting number one
Abstract: The unknotting number of a knot, $K$, is a defined to be the minimal number of crossing changes required in any diagram of $K$ to obtain the unknot. Despite its straight-forward definition, the unknotting number is frequently difficult to compute. For alternating knots it can be shown that a knot has unknotting number one if and only if every alternating diagram contains an unknotting crossing. I will explain the proof of this result, which uses an obstruction, originally due to Greene, arising from Donaldson's Theorem and Heegaard Floer homology. This approach also shows that an alternating knot has unknotting number one if and only if its double branched cover arises by half-integer surgery on a knot in $S^3$. I will also talk about how one can extend this work to show that if the branched double cover of an alternating link arises as non-integer surgery on a knot in $S^3$, then this is exhibited by a rational tangle replacement in an alternating diagram.

Frank Quinn (Virginia Tech)
Title: Some decompositions of smooth 4-manifolds
Abstract: We describe several ways to produce decompositions, including simplified versions of the Gay-Kirby trisections, the authorís dual decompositions, and descriptions as singular 2-parameter families of surfaces. These are all extremely complicated and it is not yet clear how they might be useful. Better connections with analysis are probaby needed for defining invariants. Good questions that allow some changes in the manifold might provide guidance.

Jennifer Schultens (Davis)
Title: Kakimizu complexes
Abstract: In the 1980's, Osamu Kakimizu defined a complex associated with a knot.The complex encodes isotopy classes of Seifert surfaces.  Several teams of researchers studied the complex over the intervening years.  Most notably, Johnson-Wilson-Pelayo showed that the Kakimizu complex of a knot is quasi-Euclidean.  The Kakimizu complex can be defined more generally, for 3-manifolds and also for surfaces.  I will discuss this more general definition.  I will show how the Kakimizu complex of a surface compares to and differs from the curve complex and homology curve complex.  I will exhibit examples demonstrating that the Kakimizu complex of a 3-manifold is not, in general, quasi-Euclidean.

Saso Strle (Ljubljana)
Title: Essential non-orientable surfaces in homology cobordisms and closed 4-manifolds
Abstract: I will discuss constraints on embeddings of a non-orientable surface in a given homology class in a 4-manifold. These take the form of inequalities involving the genus and normal Euler class of the surface, and some Heegaard-Floer theoretic invariants. The methods apply to embeddings in closed 4-manifolds or in a homology M ◊ I, where M is a rational homology 3-sphere. In the latter case it is interesting to compare the resulting genus bounds with those in the 3-manifold. For example, it turns out that for lens spaces the 3- and 4-dimensional bounds agree. This is joint work with Adam Levine and Daniel Ruberman.


Peter Teichner (Max Planck & UC Berkeley)

Title: Pulling apart 2-spheres in 4-space
Abstract: In joint work with Rob Schneiderman, we completed the computation of the group of link maps of two 2-spheres into 4-space, up to link homotopy. The proof uses standard techniques like the Whitney move, symmetric surgery and algebraic self-intersection invariants but in addition introduces new 4-manifold tools like Whitney spheres and constructions of accessory disks.

Abigail Thompson (UC Davis)
Title: Incompressible surfaces and distance 2 manifolds
Abstract: Let M be a closed orientable 3-manifold with minimal genus Heegaard surface F.    If the distance of the Heegaard splitting is at least 3, it is known that M is atoroidal and hence hyperbolic.    So all toroidal manifolds have distance at most 2; but not all distance 2 manifolds are toroidal.   We would like to be able to sort out which are which, and we give some results towards this end.  In particular, we show that if such an M is toroidal, M either contains a nicely-positioned incompressible surface (with respect to a minimal genus Heegaard surface), or M has a small Seifert-fibered space component in its JSJ decomposition, or the minimal genus splitting of M is not thin.    

Alexander Zupan (University of Nebraska-Lincoln)
Title: Trisections of 4-manifolds and Dehn surgery
Abstract:  A trisection of a 4-manifold X is a splitting of X into three simple pieces that glue together in a prescribed way.  Every trisection gives rise to a handle decomposition of X.  As such, it is often possible to extract results about trisections from known facts about Dehn surgery, and vice versa.  In this talk, we will explore this relationship; in particular, we will discuss connections between trisections and different versions of the Generalized Property R Conjecture.  Much of this talk is based on joint work with Jeffrey Meier and Trent Schirmer.



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