Floer homologies and topology of 4-manifoldsApril 8-9, 2017, University of Massachusetts, Amherst, MA
Organizing CommitteeR. Inanc Baykur (UMass Amherst), Weimin Chen (UMass Amherst) and Daniel Ruberman (Brandeis University)
The aim of this 2-day conference is to bring together active researchers in Floer homology and topology of 4-manifolds, and provide a panorama of the field through a variety of talks and discussions. There will be six 1-hour long talks, accompanied by four 1/2-hour long talks by PhD students, and plenty of ample time for discussions in between.
Registration and support
[Funding applications are now closed.]
Funding is available for conference participation. Support requests should be made at the registration link by March 16, 2017. We particularly encourage graduate students and recent PhDs to apply.
We have reserved rooms for all the invited speakers and participants coming from outside of Western Massachusetts at the UMass Hotel and Conference Center. Please provide your planned dates of stay and accommodation preferences at the registration link. Further inquiries regarding accommodation should be made to [wchen at math dot umass dot edu].
This event is sponsored by the National Science Foundation Grant DMS-1522633.
The closest airport to Amherst is the Bradley (Hartford/Springfield) Airport, and there are shuttle services from/to UMass Amherst. The second major airport in the area is the Logan (Boston) Airport (1,5-2hrs distance), which however requires an additional bus commute between Boston and Amherst. Greyhound buses serve between Amherst and many other locations such as Boston and New York.
Saturday, April 8
9:00- 9:30 REGISTRATION
9:30-10:30 John Baldwin
11:00-12:00 Jianfeng Lin
12:15-12:45 Henry Horton
12:45- 2:15 LUNCH
2:15- 3:15 Liam Watson
3:30- 4:30 Arunima Ray
4:30- 5:00 REFRESHMENTS
5:00- 5:30 Michael Wong
Sunday, April 9
9:00-10:00 Josh Greene
10:30-11:30 Aliakbar Daemi
11:45-12:15 Biji Wong
12:30- 1:00 Boyu Zhang
All talks will be held at Lederle Graduate Research Tower, Room 1634. Tea and
refreshments will be served in the lounge area next to LGRT 1634.
Titles and abstracts
Khovanov homology detects the trefoil
In 2010, Kronheimer and Mrowka proved that Khovanov homology detects the unknot, answering a "categorified" version of the famous open question: Does the Jones polynomial detect the unknot? An even more difficult question is: Does the Jones polynomial detects the trefoils? The goal of this talk is to outline our proof that Khovanov homology detects the trefoils, answering a "categorified" version of this second question. Our proof, like Kronheimer and Mrowka's, relies on a relationship between Khovanov homology and instanton Floer homology. More surprising, however, is that it also hinges fundamentally on several ideas from contact and symplectic geometry. This is joint work with Steven Sivek.
Instantons and homology cobordism group
The set of 3-manifolds with the same homology as the 3-dimensional sphere, modulo an equivalence relation called "homology cobordance", forms a group. The additive structure of this group is given by taking connected sum. This group is called the "homology cobordism group" and plays a special role in low dimensional topology and knot theory. In this talk, I will explain how instantons can be used to construct a family of invariants of the homology cobordism group. The relationship between these invariants and the Froyshov's celebrated invariant will be discussed. I will also talk about some topological applications.
(1,1) L-space knots
I will describe a diagrammatic classification of (1,1) knots in S^3 and lens spaces that admit non-trivial L-space surgeries. A corollary of the classification is that 1-bridge braids in these manifolds admit non-trivial L-space surgeries. This is joint work with Sam Lewallen and Faramarz Vafaee.
Symplectic instanton homology: SO(3)-bundles, functoriality, and Dehn surgery
I will describe a new construction of a Floer theoretic invariant of a 3-manifold Y equipped with a principal SO(3)-bundle ω, as well as outline some of its properties. The invariant arises as the Lagrangian Floer homology of so-called traceless character varieties associated to a Heegaard splitting of the 3-manifold; we call it the "symplectic instanton homology" of (Y, ω) and denote it SI(Y, ω). If K is a (framed) knot in Y and Y0, Y1 denote 0- and 1-surgeries on K (with respect to the given framing), we indicate how the 2-handle cobordism maps induced by these surgeries fit into an exact triangle of symplectic instanton homologies, ··· → SI(Y, ω + ωK) → SI(Y0, ω) → SI(Y1, ω) → ···, where ωK is the SO(3)-bundle on Y dual to K. Time permitting, we will indicate how for link surgeries there is more generally a spectral sequence of symplectic instanton homologies, which gives a relationship to reduced Khovanov homology when applied to the branched double cover of a link.
The Seiberg-Witten equations on end-periodic manifolds and positive scalar curvature metrics
It is an open question which 4-manifolds admit metrics of positive scalar curvature. In this talk, I will introduce a new obstruction for existence of such metric on compact 4-manifolds with the same homology as the circle cross 3-sphere. The main tool is the Seiberg-Witten equations on end-periodic manifolds. This obstruction is given in terms of the relation between the Frøyshov invariant of the generator of the third homology with the Casson-type Seiberg-Witten invariant, as defined by Mrowka-Ruberman-Saveliev.
4-dimensional analogues of Dehn's lemma
We investigate certain 4-dimensional analogues of the classical 3-dimensional Dehn's lemma, giving examples where such analogues do or do not hold, in the smooth and topological categories. For instance, we show that an essential 2-sphere S in the boundary of a simply connected 4-manifold W such that S is null-homotopic in W need not extend to an embedding of a ball in W. However, if W is simply connected (or more generally, has abelian fundamental group) with boundary a homology sphere, then S bounds a topologically embedded ball in W. Moreover, we give examples where such an S does not bound any smoothly embedded ball in W. We give similar results for tori: we construct incompressible tori T in the boundary of a contractible 4-manifold W such that T extends to a map of a solid torus in W, but not to any embedding of a solid torus in W. Moreover, we construct an incompressible torus T in the boundary of a contractible 4-manifold W such that T extends to a topological embedding of a solid torus in W but no smooth embedding. (This is joint work with Danny Ruberman.)
Bordered invariants via immersed curves
I'll describe joint a project with Jonathan Hanselman and Jake Rasmussen interpreting bordered Heegaard Floer homology for manifolds with torus boundary in terms of immersed curves. This machinery gives an intersection-theoretic version of the box tensor product underpinning the pairing theorem of Lipshitz-Ozsvath-Thurston that is essential for cut-and-paste techniques in Heegaard Floer theory. The goal of the talk will be to outline the background and motivation for, as well as some of the applications of, these new tools.
Equivariant corks and Heegaard Floer homology
A cork is a contractible smooth 4-manifold with an involution on its boundary that does not extend to a diffeomorphism of the entire manifold. Corks can be used to detect exotic structures; in fact any two smooth structures on a closed simply-connected 4-manifold are related by a cork twist. Recently, Auckly-Kim-Melvin-Ruberman showed that for any finite subgroup G of SO(4) there exists a contractible 4-manifold with an effective G-action on its boundary so that the twists associated to the non-trivial elements of G do not extend to diffeomorphisms of the entire manifold. We use Heegaard Floer techniques originating in work of Akbulut-Karakurt to give a different proof of this phenomenon.
An unoriented skein relation for tangle Floer homology
Although the Alexander polynomial does not satisfy an unoriented skein relation, Manolescu (2007) showed that knot Floer homology satisfies an unoriented skein exact triangle. In this talk, we prove that an analogous skein relation is satisfied by the Petkova-Vertesi tangle Floer homology, which together with a gluing theorem recovers a version of Manolescu's result. This is joint work with Ina Petkova.
A monopole invariant for foliations without transverse invariant measure
The question about the existence and flexibility of taut foliations on three manifolds has been studied for decades. Floer-theoretical obstructions for the existence of taut foliations on rational homology spheres have been obtained by Kronheimer, Mrowka, Ozsvath, and Szabo. Recently, Vogel and Bowden constructed examples of taut foliations that are homotopic as distributions but can not be deformed to each other through taut foliations. In this talk we will propose a different approach. Instead of perturbing the foliation to a contact structure, we directly study a symplectization of the foliation itself, which will give to a canonically defined element in the monopoe Floer homology. It turns out that this element can shed some light on the question of existence and flexibility of taut foliations.