# Interactions of Gauge Theory with Contact and Symplectic Topology in Dimensions 3 and 4 (16w5096)

## Organizers

Denis Auroux (University of California, Berkeley)

Hans Boden (McMaster University)

Olivier Collin (Université de Québec à Montréal)

John Etnyre (Georgia Institute of Technology)

## Objectives

The proposed workshop will bring together researchers working in various areas of geometric topology, symplectic and contact geometry and topology, and gauge theory, in order to foster collaborations between these different groups and explore a variety of approaches to problems in low-dimensional topology. These areas of mathematics have recently had highly fruitful interactions and are poised for more in the future. Evidence for this is provided in the discussion below which outlines the many spectacular results on problems of fundamental importance that have recently been proved by researchers working in these areas.

**Unification of invariants: ** There has been much recent 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 the monopole Floer theory with embedded contact homology, and as an application, they have succeeded in proving the Arnold chord conjecture for legendrian knots in arbitrary contact 3-manifolds. More recent research points to a convergence of Floer theories, and two separate teams, namely Kutluhan--Lee--Taubes and Colin--Ghiggini--Honda, working independently, have established the equivalence between Heegaard-Floer homology, embedded contact homology and monopole Floer homology. These are fundamentally important and very deep results, the ramifications of which are still being explored. An important open problem in this area is to understand how instanton Floer homology compares with monopole Floer and Heegaard--Floer.

There are a number of other striking results coming out of Floer theory, including a beautiful and elegant characterization of fibered knots in terms of Heegaard--Floer groups given by Y. Ni and P. Ghiggini, the establishment of a spectral sequence relating Khovanov homology to the instanton Floer knot homology with application to showing Khovanov homology detects the unknot given by P. Kronheimer and T. Mrowka, and the solution to the Weinstein conjecture in dimension 3 given by Taubes using embedded contact homology and monopole Floer homology. Taubes proved that, for any compact oriented 3-manifold $M$ with contact 1-form $alpha$, the vector field that generates the kernel of the 2-form $dalpha$ has at least one closed integral curve. Another very exciting and fundamental development is the solution to the triangulation conjecture given by C. Manolescu using Pin(2) Floer homology. The triangulation conjecture is the statement that every compact topological manifold can be triangulated by a locally finite simplicial complex, and this was known to be false in dimension four but remained an open problem for manifolds of dimension five and higher. In the early 1980s, Galewski and Stern had shown the triangulation conjecture to be equivalent to the statement that there exists a homology 3-sphere of Rochlin invariant one with order two in the homology cobordism group $Theta^3.$ In a recently posted article, Manolescu develops Pin(2) Floer homology theory for homology 3-spheres and uses it to disprove the triangulation conjecture in dimension five and higher by showing such homology 3-spheres cannot exist.

Another important goal is to understand the relationship between the various Floer-type invariants 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 (rather than geometric). Ozsv'ath and Szab'o showed that Khovanov's homology of a link is related to the Heegaard Floer homology of its double branched cover by a spectral sequence. The progress accomplished on 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 should remain an active area of research for the coming years, as it also relates to the link invariants constructed by Seidel and Smith using the symplectic geometry of nilpotent slices, and recent progress in understanding the structure of Fukaya categories suggests new lines of approach. In another direction, building on their foundational results on singular instantons, Kronheimer and Mrowka have established the existence of a spectral sequence from Khovanov homology to the $SU(2)$ instanton knot homology, and consequently they show that the Khovanov homology detects the unknot. Their work suggests an intriguing relationship between Khovanov-Rozansky homology and $SU(n)$ instanton knot homology, and Witten's work recasting Khovanov homology in terms of 4-dimensional gauge theory, while not yet well understood by mathematicians, most likely will serve as a catalyst for other exciting new developments.

**TQFTs and the algebraic structure of invariants: ** It appears that many invariants of 3-manifolds or links can be extended to invariants of 3-manifolds with boundary or tangles, forming ``extended topological field theories''. While this was a built-in feature of Khovanov homology, the discovery of similar structures in Floer-type invariants is more recent. While Juh'asz's ``sutured'' Heegaard-Floer homology already led to exciting applications (such as detecting fiberedness), the recent introduction by Lipshitz, Ozsv'ath and Thurston of ``bordered Floer homology'' has led to a much richer algebraic picture of Heegaard-Floer theory. This not only opens new perspectives for computations (see below), but also provides insight into the structure of Heegaard-Floer theory and its relation to Khovanov homology. Recent work of Douglas and Manolescu suggests that the story extends even further, to surfaces with boundary and 3-manifolds with corners. Bordered and sutured versions of other Floer-type invariants have not yet been as thoroughly developed, but appear full of promise and should be a hotbed of future activity; for instance, sutured instanton Floer homology plays a key role in Kronheimer and Mrowka's work mentioned above. In a different direction, recent progress on algebraic structures in Fukaya categories and Legendrian contact homology (by Bourgeois, Ekholm, Eliashberg; Abouzaid, Seidel; Ganatra) is likely to have significant applications in low-dimensional topology.

**Developing computational techniques: ** Most of the invariants arising from gauge theory and contact / symplectic topology involve spaces of solutions to geometric PDEs, which makes explicit computations particularly difficult. In the past few years there has been dramatic progress in several directions. The problem of combinatorially constructing Heegaard-Floer groups without 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. While combinatorial Heegaard-Floer homology continues to develop at a spectacular pace, Lipshitz, Ozsv'ath and Thurston's bordered Floer homology offers a different approach to computing Heegaard-Floer homology by decomposing a 3-manifold into a sequence of elementary cobordisms between oriented surfaces. In a different direction, a result of Bourgeois, Ekholm and Eliashberg gives a way to compute the contact homology of a contact manifold obtained from another one by Legendrian surgery. 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 in terms of its Legendrian surgery description. With recent progress on the classification of Legendrian knots this could yield a flood of information about contact 3-manifolds.

**Exploiting interactions between constructions and invariants: ** The emergence of invariants of embeddings from contact homology is another very promising avenue of research. 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. The contact homology of this Legendrian gives an invariant of the original embedding. Ekholm, Etnyre, Ng and Sullivan have recently given a rigorous computation of this invariant for knots in 3-space and shown it is equal to a very powerful combinatorial invariant defined by Ng. This new invariant has surprising connections with many classical knot invariants and seems quite strong; it also appears to give new information about transverse knots in contact manifolds. Exploring its properties 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. In spite of recent progress by Bourgeois, Ekholm and Eliashberg, there is still much work to do to extract computable information that can be used in applications. In the end though, it is expected that the theory will be invaluable and provide more invariants, not only for Legendrian knots in contact 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, and further results of his about exact Lagrangians in cotangent bundles. Can such ideas be exploited in dimension 4 to attack the smooth Poincar'e conjecture?

Indeed, one of the driving open problems in 4-dimensional topology is the smooth Poincar'e conjecture. Recent work of Freedman, Gompf, Morrison and Walker reveals that Khovanov homology can be used to give an obstruction to specific handle decompositions of homotopy 4-spheres being the actual 4-sphere. This development sparked considerable interest as a method for identifying counterexamples to the smooth 4-d Poincar'e conjecture, assuming they exist! Akbulut and Gompf subsequently proved 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 4-manifolds that are as ``small'' as possible. After Freedman and Donaldson's work in the early 1980's gave the first examples of exotic smooth structures, and Kotschick's result for $\mathbb{CP}^2 \overline{\mathbb{CP}}{}^2$, there was little progress until J. Park's breakthrough in 2005. 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 or rim surgery or Luttinger surgery along particularly well-chosen embedded surfaces, together with an intimate understanding of their effect on Seiberg-Witten invariants, one can often deduce the presence of several (generally infinitely many) exotic smooth structures (only some of which carry symplectic forms). It is reasonable to expect further progress on this important problem for other small symplectic 4-manifolds (e.g. $\mathbb{CP}^2 \overline{\mathbb{CP}}{}^2$ or $S^2times S^2$) via this approach.

**Contact structures on 3-manifolds and Heegaard-Floer theory: ** The existence of tight contact structures on 3-manifolds has been an important subject of investigation for a long time, and one on which significant progress has been made in the last decade. 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 incremental steps by a number of 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 non-vanishing criterion for the contact invariant of Ozsv'ath and Szab'o. 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. They have also defined fractional Dehn twist coefficients (FDTC) for diffeomorphisms of surfaces with boundary. Large FDTC frequently implies an associated contact manifold is tight. Moreover, recently Hedden and Mark have used Heegaard-Floer theory to show that FDTCs are bounded on a given manifold. In another direction Etnyre, Vela-Vick and Zarev have shown how to use contact geometry to recover most flavors of knot Heegaard-Floer theory from sutured Floer theory. These works show intricate connections between Heegaard-Floer theory and contact geometry. It is hoped that as we develop a better understanding of the relations between fundamental groups and Heegaard-Floer theory we will be able to use some of these connections to illuminate the questions of tight contact structures on hyperbolic manifolds.

**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), March 2009 (Interactions of geometry and topology in dimensions 3 and 4: BIRS 09w5095), March 2011 (Interactions between contact/symplectic topology and gauge theory in dimensions 3 and 4: BIRS 11w5085), and March 2013 (Interactions of gauge theory with contact and symplectic topology in dimensions 3 and 4: BIRS 13w5037). These events were hugely successful in every respect, as discussions among participants were not only informative but also led to new results and fruitful collaborations. All four previous workshops had waiting lists of people wishing to attend, and we expect interest in the currently proposed workshop to be just as high. Indeed, the wealth of recent results in the field shows that this area of mathematics is developing at a very fast pace, and we are very confident that the progress over the next two years will lead to another 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 be placed on including post-doctoral researchers and current Ph.D. students. The schedule will include five 50-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.