Interactions between contact symplectic topology and gauge theory in dimensions 3 and 4 (11w5085)

Arriving in Banff, Alberta Sunday, March 20 and departing Friday March 25, 2011


Denis Auroux (University of California, Berkeley)

(McMaster University)

Olivier Collin (Université du Quebec à Montréal)

(Georgia Institute of Technology)


Over the last several decades it has become clear that the topology of manifolds in low-dimensions is subtly and beautifully intertwined with various flavors of geometry, such as hyperbolic, symplectic and contact, as well as with ideas from physics, such as gauge theories and mirror symmetry. Collaborations among people working in these diverse areas has exploded over the last few years resulting in the solutions to venerable conjectures in topology as well as the birth of entire new sub-fields and perspectives in these areas.
Highlights of some of the more spectacular recent results include the characterization of which 3--manifolds admit a symplectic structure when crossed with $S^1,$ the Heegaard-Floer characterization of fibered knots, the proof of Property $P$ for nontrivial knots in $S^3,$ the solution to the Weinstein conjecture (and generalizations of it), and a deepening of our understanding of exotic smooth and symplectic structures on 4-manifolds.
Critical to these developments has been the information provided by a vast array of new invariants whose definitions were motivated by gauge theory and topological quantum field theory.
These invariants -- Donaldson-Floer,Seiberg-Witten, Ozsv'ath-Szab'o, Khovanov homology or Embedded Contact Homology to name a few -- have intriguing relations among them, and a better understanding of these will lead to significant progress not only in topology but also in contact and symplectic geometry and physics.
An even more promising direction is the interplay between these invariants and more constructive approaches to low-dimensional manifolds -- open book decompositions of contact 3-manifolds, symplectic fillings, Lefschetz fibrations, knot surgery constructions among many others. This interaction between powerful invariants and constructive methods is now more than ever a major driving force in the subject. Below we will survey some of the most active branches of low-dimensional topology, thereby outlining natural directions and objectives for the workshop.

Unification of invariants:
Recently there has been much progress in showing that various invariants defined in starkly different ways actually compute the same thing. This has allowed for many striking results. For example, Taubes and Hutchings have a program for identifying Seiberg-Witten Floer theory with Embedded Contact Homology, and using these ideas, Taubes succeeded in proving the 3-dimensional Weinstein Conjecture: for any compact oriented 3-manifold $M$ and $alpha$ a contact 1-form on $M$, the vector field that generates the kernel of the 2-form $dalpha$ has at least one closed integral curve. Further developments have allowed for extensions and refinements of the Weinstein conjecture and it appears we are on the cusp of identifying the two theories. The ramifications of such a convergence of theories is as yet unknown but given the spectacular results stemming from earlier work, one expects great things.
Similarly, progress on the Pidstrigatch and Tyurin program to prove the Witten conjecture relating instanton Floer homology with Seiberg-Witten Floer homology was a crucial ingredient in the solutions of the famous conjecture that all non-trivial knots in $S^3$ have Property P: that is that non-trivial surgery yields a manifold with non-trivial fundamental group. More recently Kronheimer and Mrowka simplified their proof and, in fact, proved stronger results by adapting work of Juh'asz on sutured Heegaard-Floer homology to the context of instanton Floer homology, another good indication that unified invariants tend to yield deep results in the area.

Another current trend in the area is the understanding of the relationship
between the various invariants of Floer type for knots and 3-manifolds and
Khovanov homology. Khovanov homology was constructed as a categorification of the Jones
polynomial of knots and its nature is very algebraic and combinatorial.
Ozsv'ath and Szab'o derived a spectral sequence whose $E^2$ term is a suitable variant of Khovanov's homology for a link, converging to the Heegaard Floer homology of the double branched cover of the link. The progress described above in combinatorial Heegaard-Floer homology has already enabled Manolescu and Ozsv'ath to explore further the relationship between the two theories, through the notion of homological thinness. This is an active line of research that could also tie in to the link invariant that Seidel and Smith constructed using the symplectic geometry of nilpotent slices. In another direction, current work of Kronheimer and Mrowka, going back to their foundational work on singular instanton connections over 4-manifolds, seems to suggest another relationship between Khovanov homology and the original instanton Floer homology developed almost 20 years ago. Since the various Floer theories for knots detect the unknot, it might be possible that such delevopments culminate in a proof that Khovanov homology (and the Jones polynomial) detect the unknot.

Developing computational techniques:
Most of the topological invariants arising from gauge theory and contact / symplectic topology rely extensively on analytical tools, which makes explicit computations particularly difficult since information about
spaces of solutions to such PDE problems is scarce. In the past few years there
has been dramatic progress in combinatorial approaches to Ozsv'ath-Szab'o
theory as well as Contact Homology. Indeed, the problem of combinatorially constructing Heegaard-Floer groups
without resorting to counting pseudo-holomorphic curves has taken a very
promising turn as knot Floer homology was given a purely combinatorial
interpretation by Manolescu, Ozsv'ath and Sarkar. This has already led to
progress in the classification of transverse knots in contact manifolds as well as work by Ng on bounds for the Thurston-Bennequin invariant of Legendrian
knots. It is expected that the theory will progress greatly over the course
of the next few years thanks to the combinatorial set-up. Moreover, Bourgeois, Ekholm and Eliashberg have constructed an exact sequence that allows one to compute the contact homology of a contact manifold obtained from "Legendrian surgery" on another one. This construction is particularly "simple" in dimension 3 where there is essentially an algorithm for writing down the contact homology of a contact 3-manifold obtained from Legendrian surgery on a Legendrian knot. With recent progress on the classification of Legendrian knots in various knots types this could yield a flood of information about contact 3-manifolds.

Very recently, Lipshitz, Ozsv'ath and Thurston have opened a whole new direction by extending Heegaard-Floer homology to the case of 3-manifolds with boundary. Among other applications, this allows one to compute Heegaard-Floer homology by decomposing a 3-manifold into a sequence of elementary cobordisms between oriented surfaces.

Exploiting interactions between constructions and invariants:
The emergence of invariants of embeddings from contact homology is another one of the very promising avenues of research in the area. Given a manifold embedded
in Euclidean space, one can look at its unit conormal bundle in the unit
cotangent bundle of Euclidean space to get a Legendrian submanifold.
Computing the contact homology of this Legendrian gives an invariant of the
original embedding. Recent work of Ekholm, Etnyre, Ng and Sullivan rigorously
computes this invariant for knots in 3-space and shows it is equal to a very
powerful combinatorial invariant defined by Ng. This invariant has
surprising connections with many classical invariants of knots and seems
quite strong. Exploring this new invariant of knots and extending it to
other situations should be a fruitful line of research for years to come.
Moreover, contact homology is only the tip of the iceberg of Symplectic Field
Theory (SFT). This theory, introduced by Eliashberg, Givental and Hofer, has been an inspirational and driving force in symplectic geometry for over a decade
now, and recent advances in its rigorous definition suggest that a precise
formulation of the relative version will emerge in the coming years. It
appears there will still be much work to do to extract computable and meaningful pieces that one can use in applications. In the end though, it is
expected that the theory will be invaluable in symplectic and contact geometry
and will provide more invariants, not only for Legendrian knots in contact
3-manifolds and Lagrangian cobordisms between them, but also for topological
knots by considering the conormal construction mentioned above. Evidence for this comes from Abouzaid's recent demonstration that the symplectic geometry of cotangent bundles can be used to distinguish exotic smooth structures on spheres of high dimension. Can such ideas be exploited in dimension 4 to attack the smooth Poincar'e conjecture?

One of the driving questions in 4-dimensional topology is the smooth Poincar'e conjecture and its symplectic analog.It is rather unbelievable that topologists still don't know how many smooth structures there are on the 4-sphere or the complex projective 2-space, and which admit symplectic structures. There has recently been a burst of activity in this area. Freedman, Gompf, Morrison and Walker have shown how to use Khovanov homology to get an obstruction to specific handle decompositions of homotopy 4-spheres being the actual 4-sphere (that is this obstruction could identify a counterexample to the smooth Poincar'e conjecture, if it exists!). After this work Akbulut and Gompf showed that many potential counterexamples to the Poincar'e conjecture are actually the standard sphere. Another approach to such problems is to try to build exotic smooth structures on "smaller and smaller" 4-manifolds. After Freedman and Donaldson's work in the early 1980s the problem for $CP^2$ # ${}_noverline{CP^2}$ could be tackled for $n=9$, Kotschick handled the case $n=8$, but there was little progress made until J. Park's breakthrough a few years ago. There has since been a flurry of activity on existence of exotic smooth structures on small symplectic 4-manifolds by different teams of researchers (Akhmedov-Park, Baldridge-Kirk, and Fintushel-Stern-Park). The advances are made by exploiting a certain tension between constructions and invariants. Using clever new cut-and-paste constructions such as knot and rim surgery, together with an intimate understanding of their effect on invariants such as the Seiberg-Witten invariants, one can often deduce the presence of several (generally infinitely many) exotic smooth structures. The constructions ideally involve modifying the 4-manifold so as to alter the invariants without destroying the symplectic structure or homeomorphism type. This requires one to perform surgeries along a particularly well-chosen surface embedded in the 4-manifold. It is reasonable to expect further progress on this important problem for other small symplectic 4-manifolds (e.g. ${mathbb C}{mathbb P}^2$, ${mathbb C}{mathbb P}^2$ # ${}overline{{mathbb C}{mathbb P}}^2$, or $S^2times S^2$ via the various approaches that have been developed and the continued influence of the powerful 4-manifold invariants arising from gauge theory and symplectic geometry.

Relating contact structure, Heegaard-Floer theory and 3-manifolds:
The existence of tight contact structures on 3-manifolds has been an important subject of investigation for a long time and, since the year 2000, significant progress has been made in our understanding of which 3-manifolds admit tight contact structures. This fundamental question has potential applications not only to contact geometry but also to low-dimensional
topology and dynamics. It also illustrates very well the natural interactions between the invariants described above and constructive methods. After many incremental steps by several mathematicians, Lisca and Stipsicz have completely classified which Seifert fibered 3-manifolds admit a tight contact structure. Their approach relies heavily on Heegaard-Floer homology through a on-vanishing criterion for the contact invariant of Ozsvath and Szabo for Seifert fibered manifolds. On the other hand, geometric methods reminiscent of the theory of normal surfaces of Haken and Kneser have enabled Colin, Giroux and Honda to establish general results such as: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures. One of the outstanding and fundamental questions here is the understanding of tight contact structures on hyperbolic 3-manifolds. Work of Kazez, Honda and Mati'c has led to a characterization of tight 3-manifolds in terms of right-veering diffeomorphisms, a step which should make calculations in contact homology and Heegaard Floer homology manageable, but thus far the condition of a manifold being hyperbolic has not been properly understood in this context. It is hoped that the current wide-ranging technology will help elucidate the problem of tight structures on 3-manifolds.

Statement of Objectives

The proposed workshop is a follow-up event to the well received BIRS workshops we organized in March 2007 (Interactions of Geometry and Topology in Low Dimensions : BIRS 07w5033) and March 2009 (Interactions of Geometry and Topology in dimensions 3 and 4: 09w5095). These events were hugely successful in many respects, as discussions among participants were not only informative but also led to new results and fruitful collaborations. Both previous workshops had waiting lists of people wishing to attend and we expect interest in the currently proposed workshop to be even greater. The varied themes in the proposal show how the area is developing at a steady pace and we are very confident that the progress over the next two years will lead to an important and productive meeting.

The workshop will bring together a diverse group of mathematicians. Among the invited participants are experts working in 3- and 4-manifold topology and their invariants arising from gauge theory, contact and symplectic topology. While some of the leading experts in each area are expected to attend the workshop, a special emphasis will also be put on the participation of
post-doctoral researchers as well as some current Ph.D. students. The schedule will include five to six $45$ minute talks each day -- except for Wednesday where only half a day will be scheduled -- with time in the late afternoons and evenings left open for informal discussions, collaborations, and problem sessions.