# Low Dimensional Contact Topology

Project Outline

1. Workshop. We plan to invite five to eight people who are making significant contributions to contact geometry to AIM and Stanford University from August 21 to December 21 of 2000. Together with approximately ten people at Stanford University, or in the Bay Area, we shall run several weekly seminars and mini-courses. There shall be ample time for more informal discussions and collaborations. We shall also have eight to ten people visiting for shorter periods of time.
2. Conference. In December we shall hold a week long conference to publicize our work and stimulate further work in the field.
Background and Project Description

Contact geometry is an odd dimensional analog of symplectic geometry. Specifically, if M is an oriented 2n+1-manifold then an oriented contact structure \xi is a 2n-dimensional distribution (sub-bundle of the tangent bundle) for which there is a 1-form \alpha for which \xi=\hbox{ker}(\alpha) and \alpha\wedge(d\alpha)^n\not=0.

Contact structures first appeared in an explicit form in the work of Sophus Lie, and later were intensively studied by E. Cartan, Darboux, and others. The first result of topological flavor, existence of contact structures on all orientable 3-manifolds, was first obtained by Martinet [35] in 1971. The birth of symplectic topology in eighties (see [3, 29, 7, 21], et al.), greatly influenced by Arnold's conjectures (see [2]) brought an exciting development in contact topology as well. However, this odd-dimensional cousin of symplectic topology did not get as much attention as its symplectic counterpart. We hope that this program, besides its immediate mathematical consequences, will help to attract more attention to this exciting and important area.

1. Contact Geometry in dimension three.

In 3 dimensions, there are two fundamentally different types of contact structures: tight and overtwisted. This dichotomy, first observed in [11], grew out of Bennequin's study [3] of transversal knots (knots transverse to the contact planes). Bennequin proved that the contact structures on R3 defined by

\alpha_1=dz + r^2\, d\theta,
and
\alpha_2=\cos r\, dz + r\sin r\, d\theta,
are not the same by showing they support different transversal knots. The contact structure \xi_2=\hbox{ker}(\alpha_2) is the first example of an exotic'' (or non-standard) contact structure on R3. It is also the prototypical example of an overtwisted contact structure. In order to define overtwisted, note that if \Sigma is a surface in a contact manifold (M,\xi) then T\Sigma\cap \xi is a singular line field on \Sigma. We can integrate this line field to a singular foliation, \Sigma_\xi, called the characteristic foliation. Now \xi is called overtwisted if there is an embedded disk D in M such that D_\xi contains a closed leaf. Note the disk in the xy-plane with radius \pi is an overtwisted disk in for \xi_2. If \xi is not overtwisted we say it is tight. It turns out that most of the contact structures that Martinet constructed are overtwisted. Later, it was shown in [8] that it is easy to understand overtwisted contact structures: the isotopy classification of overtwisted structures is equivalent to the homotopy classification of plane fields. Moreover through the work of many people (for example see [11, 34]) it is now clear that tight contact structures have a more subtle geometric nature and are intimately related to the topology of the manifold on which they live. Thus one would like to know if all 3-manifolds admit a tight contact structure. In the recent paper [19] it was shown there are reducible 3-manifolds that do not admit tight contact structures. This prompts the following question:
Does any irreducible 3-manifold admit a tight contact structure with one orientation or another?
We hope that the answer to this fundamental question will be given as one of the results of the program.

The contact structure \xi on M is said to be filled by a symplectic manifold (X,\omega) if \partial X=M (as oriented manifolds) and \omega\vert_\xi is a symplectic form on \xi. Gromov [29] and Eliashberg [10] have used the theory of holomorphic curves in X to show that a fillable contact structure is tight. With this result one can constructed many tight contact structures on 3-manifolds (see [9, 12, 28]). Since many (maybe most) currently known tight contact structures come from this construction one is prompted to ask:

Are all tight contact manifolds symplectically fillable?

Either answer to this question would be serious progress in understanding the relation between 3 dimensional contact and the 4 dimensional symplectic topology. Providing an answer to this question is a focal point of this project. The only other general construction of tight contact structures is due to Eliashberg and Thurston [16] who have shown how to perturb a Reebless foliation into a tight contact structure. Some of the tight structures constructed this way should shed light on the above question.

Ultimately, one would like to:

Classify tight contact structures on 3-manifolds.
Through the work of many mathematicians we understand tight structures on many simple manifolds. Specifically they are classified on the 3-ball, the 3-sphere [11], lens spaces [18, 25, 32], torical annuli, solid tori, torus bundles over S1 and circle bundles over surfaces [25, 32]. We believe that the classification of tight contact structures will soon be extended to Seifert fibered spaces and possibly Haken manifolds.

2. Symplectic Field Theory.

Though there are many new techniques for understanding contact structures, the most striking is contact homology (see [13]) or its still evolving generalization, symplectic field theory. Not only are these powerful new invariants of contact structures but they also demonstrate an intimate connection between contact geometry, Hamiltonian dynamics and Gromov-Witten invariants. Contact homology is a systematic way to bring Gromov's very successful theory of pseudoholomorphic curves [29] in symplectic manifolds into the arena of contact topology. Given a contact manifold (M,\xi) one may form its symplectization (W,\omega)=(M\times R, d(e^t\alpha)) where \alpha is a contact form for \xi. In W one now considers pseudoholomorphic curves with respect to a special almost complex structure. Since W is noncompact, Gromov compactness'' of the space of pseudoholomorphic curves does not hold but the way in which it fails can be understood. In particular, compactness fails essentially because a sequence of surfaces may limit to a surface with extra punctures, and near each of these punctures the surface is asymptotic to \gamma\times R where \gamma is a closed orbit in the Reeb vector field of \alpha (recall the Reeb vector field X is uniquely defined by \alpha(X)=1 and d\alpha(X,\cdot)=0). Thus understanding the periodic orbits of the Reeb field allows one to understand the compactness issues for pseudoholomorphic curves in (W,\omega) and vice versa. For example Hofer [30] used these ideas to show, among other things, that any Reeb vector field on S3 must have a closed orbit. The contact homology of a contact manifold (M,\xi), (M a homology sphere) is formed from a chain complex which is a graded super commutative algebra over C generated by the periodic orbits of a Reeb field for \xi and a boundary map which is constructed by considering rational pseudoholomorphic curves in (W,\omega) that limit in various ways to the periodic orbits. When M is not a homology sphere then there are more complicating features. It is currently difficult to compute the contact homology for specific contact structures on 3-manifolds. As part of this project we shall:

Develop methods to compute the contact homology of contact structures on 3-manifolds.
These new methods should lead to better understanding of tight contact geometry in 3-dimensions.

3. Legendrian and transversal knots.

In trying to understand contact structures it is useful to study transversal knots (ones transversal to the contact planes) and Legendrian knots (ones tangent to the contact planes). As mentioned above, the genesis of the tight vs.\ overtwisted dichotomy was through the study of transversal knots [3]. Moreover, transversal and Legendrian knots have been used to distinguish tight contact structures on the 3-torus [33] and homology spheres [1]. The study of these special knots also provide an interesting and rich theory in their own right. Though the importance of these knots is evident, there is little known about them. In [12, 14] it was shown that simple invariants can be used to classify transversal and Legendrian unknots, while in [20] the same was shown for torus knots and the figure eight knot. In contrast, Chekanov [4] has shown that there are knots that cannot be classified in such a simple way. During this project we shall:

Extend this understanding of transversal and Legendrian knots.
To prove his above mentioned result Chekanov used a simple combinatorial version of contact homology. Eliashberg and Hofer have shown how to define a relative contact homology for pairs ((M2n+1,\xi),Ln) where L is a Legendrian submanifold, which appears to be a powerful new invariant of Legendrian knots in dimension three. We plan to:
Find effective methods to calculate the relative contact homology and better understand what it has to say about Legendrian knots.
There is an interesting conjecture of Arnold, the chord conjecture,'' that would provide some insight into the contact homology of a pair. The conjecture is:
There is a Reeb orbit connecting two points on any Legendrian knot.

4. Contact geometry in dimension five.

Not much is known concerning contact structures on 5-manifolds, and we hope that as a result of this program, the situation shall be corrected. With the new invariants of contact homology and symplectic field theory we hope to lay a foundation for the theory of contact structures on 5-manifolds. One of promising results along these lines was recently obtained by I. Ustilovsky [36]. He showed that there are infinitely many non-isomorphic contact structures on R5, which are standard at infinity, and homotopically standard. An important class of contact 5-manifolds consists of pre-quantizations of symplectic 4-manifolds with an integral cohomology class of the symplectic form. It seems likely that the symplectic field theory of the prequantization space remembers information encoded in Gromov-Witten invariants of the original manifold. This is one of the problems which, we hope, will be resolved as a result of the program.

We also hope that progress will be achieved in the problem of existence of contact structures on 5-dimensional manifolds. In 1991, Geiges [23] showed for simply connected 5-manifolds that the vanishing of the 3-rd Stieffel--Whitney class W3(M), which is a necessary homotopical condition for the existence of a contact structure on a 5-manifold M, is also a sufficient condition. For more general fundamental groups little is known (though Geiges has made some progress).

Those interested in learning more about contact geometry can consult the references sprinkled through the text above or check out our annotated suggested reading list.