Interactions of Geometry and Topology in Low Dimensions (07w5033)

Objectives

In recent years collaborations between contact and symplectic geometers, gauge theorists, and low-dimensional topologists have been highly fruitful, leading to solutions to long-standing conjectures in topology, illuminating the world of contact and symplectic manifolds, and providing new perspectives on fundamental questions in low-dimensional topology.

For some time, gauge theory has provided geometric topologists with powerful techniques yielding spectacular results on the classification problem for 4-dimensional manifolds, including the early 4-manifold invariants of Donaldson, the Seiberg-Witten invariants, and the more recent invariants of Oszvath-Szabo. In each case, when applied to 4-manifolds with boundary, the invariants take values in the Floer homology groups of the bounding 3-manifold. These Floer homology groups are important 3-manifold invariants in their own right and have been applied to a variety of problems, such as knot theory and the structure of the homology cobordism group in dimension three. Frequently, these invariants are easier to compute in the presence of extra structure. For example, all the above mentioned invariants are non-zero for closed symplectic 4-manifolds. These non-vanishing results are instrumental in the interactions between contact and symplectic geometry and low-dimensional topology.

In 2001 Peter Ozsvath and Zoltan Szabo introduced the Heegaard Floer homology groups. Their theory is motivated by Seiberg-Witten theory but defined in a completely different way. It is conjecturally equivalent to Seiberg-Witten Floer homology by an analogue of the Atyiah-Floer conjecture, and a very interesting research program of YiJen Lee proposes an approach for proving this conjecture. Heegaard Floer homology is also (conjecturally) closely related to the contact homology groups introduced by Yasha Eliashberg, Helmut Hofer and others. A good explanation of this correspondence will help determine exactly what exactly Heegaard Floer groups measure. Further, the knot Heegaard Floer homology gives a categorification of the Alexander polynomial that is formally similar to Khovanov's categorification of the Jones polynomial. One fundamental unresolved question is whether the Heegaard Floer homology groups can be defined combinatorially without resorting to counting pseudo-holomorphic curves. This is one of the problems we hope will be explored at the workshop.

Emmanuel Giroux recently revolutionized contact geometry by proving an equivalence between contact structures on 3-manifolds and open book decompositions (up to some equivalence). Open book decompositions are a classical topological concept and have been studied for some time. This result is analogous to Simon Donaldson's proof that symplectic 4-manifolds always admit Lefschetz pencils, and the analogy is strengthened by Bob Gompf's proof that all Lefschetz pencils admit symplectic structures. Both these correspondences relate geometric concepts to topological ones and have been the foundation for many of the applications of symplectic geometry to questions in low-dimensional topology. For example, this correspondence leads to two notable results, namely Giroux and Noah Goodman's positive resolution of Harer's conjecture that all fibered knots in $S^3$ are related by Hopf plumbings, and Ozsvath and Szabo's proof of Gordon's conjecture that the unknot is the only knot on which $p$ surgery yields the lens space $-L(p,1)$. The correspondence also has implications in the other directions as well. It is the basis of the non-vanishing of the Ozsvath-Szabo invariant of symplectic 4-manifolds mentioned above. It is also the key tool in Eliashberg and John Etnyre's proof that any symplectic filling of a contact manifold can be embedded in a closed symplectic manifold. This result in turn is an integral part of Peter Kronheimer and Tom Mrowka's proof of the Property P conjecture that a nontrivial surgery on a nontrivial knot in $S^3$ has nontrivial fundamental group.

The purpose of this workshop is to gather gauge theorists, contact topologists, and symplectic geometers all together to share their insights and also to foster collaborations between the fields. As mentioned above these collaborations have been highly successful in yielding important new results. For example, the 2003 BIRS Hot Topics workshop led to Kronheimer and Mrowka's solution to the Property P conjecture discussed above. We hope to spark further progress along similar lines.

The workshop will bring together a diverse group of mathematicians. Among the invited participants are experts working in 3-and 4-manifold topology, gauge theory, and contact and symplectic topology. In addition we will invite quite a few recent Ph.D.s in these areas. Because of the successful interplay between these fields, we believe this workshop has potential for forging new and fruitful colaborations. The schedule will include five $45$ minute talks each day, with time in the late afternoons and evenings left open for informal discussions, collaborations, and problem sessions.