Floer homologies and topology of 4manifolds
April 89, 2017, University of Massachusetts, Amherst, MA
Organizing Committee
R. Inanc Baykur (UMass Amherst), Weimin Chen (UMass Amherst) and Daniel Ruberman (Brandeis University)
Invited speakers


Goals
The aim of this 2day conference is to bring together active researchers in Floer homology and topology of 4manifolds, and provide a panorama of the field through a variety of talks and discussions. There will be six 1hour long talks, accompanied by four 1/2hour 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 DMS1522633.
Travel
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 (1,52hrs 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.
Schedule
Saturday, April 8 9:00 9:30 REGISTRATION 9:3010:30 John Baldwin 10:3011:00 TEA 11:0012:00 Jianfeng Lin 12:1512:45 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 9:0010:00 Josh Greene 10:0010:30 TEA 10:3011:30 Aliakbar Daemi 11:4512:15 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
John Baldwin
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.
Aliakbar Daemi
Instantons and homology cobordism group
The set of 3manifolds with the same homology as the 3dimensional 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.
Josh Greene
(1,1) Lspace knots
I will describe a diagrammatic classification of (1,1) knots in S^3 and lens spaces that admit nontrivial Lspace surgeries. A corollary of the
classification is that 1bridge braids in these manifolds admit nontrivial Lspace surgeries. This is joint work with Sam Lewallen and Faramarz Vafaee.
Henry Horton
Symplectic instanton homology: SO(3)bundles, functoriality, and Dehn surgery
I will describe a new construction of a Floer theoretic invariant
of a 3manifold 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 socalled traceless character varieties associated to a
Heegaard splitting of the 3manifold; 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 Y_{0}, Y_{1} denote 0 and 1surgeries on K
(with respect to the given framing), we indicate how the 2handle cobordism
maps induced by these surgeries fit into an exact triangle of symplectic
instanton homologies, ··· → SI(Y, ω +
ω_{K}) → SI(Y_{0}, ω) →
SI(Y_{1}, ω) → ···, where
ω_{K} 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.
Jianfeng Lin
The SeibergWitten equations on endperiodic manifolds and positive scalar curvature metrics
It is an open question which 4manifolds admit metrics of positive scalar curvature. In this talk, I will introduce a new obstruction for existence
of such metric on compact 4manifolds with the same homology as the circle cross 3sphere. The main tool is the SeibergWitten equations on endperiodic
manifolds. This obstruction is given in terms of the relation between the Frøyshov invariant of the generator of the third homology with the Cassontype
SeibergWitten invariant, as defined by MrowkaRubermanSaveliev.
Arunima Ray
4dimensional analogues of Dehn's lemma
We investigate certain 4dimensional analogues of the classical 3dimensional 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 2sphere S in the boundary of a simply connected 4manifold W such that
S is nullhomotopic 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 4manifold 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 4manifold 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.)
Liam Watson
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 intersectiontheoretic version of
the box tensor product underpinning the pairing theorem of LipshitzOzsvathThurston
that is essential for cutandpaste 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.
Biji Wong
Equivariant corks and Heegaard Floer homology
A cork is a contractible smooth 4manifold 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 simplyconnected 4manifold are related by a cork twist. Recently, AucklyKimMelvinRuberman showed that for any finite subgroup G of SO(4) there exists a contractible 4manifold with an effective Gaction on its boundary so that the twists associated to the nontrivial elements of G do not extend to diffeomorphisms of the entire manifold. We use Heegaard Floer techniques originating in work of AkbulutKarakurt to give a different proof of this phenomenon.
Michael Wong
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 PetkovaVertesi tangle Floer homology, which
together with a gluing theorem recovers a version of Manolescu's result. This is joint work with Ina Petkova.
Boyu Zhang
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.
Floertheoretical 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.