Contact Geometry, Contact Homology and Open Book Decompositions

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Abstract Award: DMS-0804820 Principal Investigator: John B. Etnyre The focus of this proposal is to better understand the intriguing nature of contact structures in all (odd) dimensions, with special attention given to dimension three and to apply contact geometric techniques to questions in topology. The first main theme of the proposed research is open book decompositions. The connection of contact geometry and these specially fibered links have provided a fount of information concerning contact structures and low-dimensional topology. As part of this proposal this fundamental connection will be exploited to define and study new invariants of contact structures and special knots in them - specifically, Legendrian and transversal knots. From prior work it is clear these invariants have subtle connections to the famed tight vs. overtwisted dichotomy in dimension three and properties of symplectic fillings of contact structures, which in turn are crucial to applications to topology. Some of the major outcomes expected from this part of the project are various classification results for Legendrian and transversal knots in tight and overtwisted contact structures and a better understanding of the fundamental process of contact surgery. The project will also study the connection between contact structures and open books in high dimensions with the goal of illuminating existence questions in high dimensions and trying to generalize the notion of overtwistedness here as well. Similar to the connection with open books, contact structures are also related to foliations. This connection has been instrumental in applications of contact geometry to low-dimensional topology. Many of the outstanding questions concerning this connection will be explored. A second main theme of the proposal is the development and analysis of Legendrian contact homology in higher dimensions. There is a great deal of beautiful analytic and algebraic structure in this theory, much of which is yet to be discovered. Moreover, through the elegant conormal construction one can then use this to define invariants of manifolds and of their embeddings in Euclidean space. These invariants appear to be extremely powerful and will be thoroughly investigated. Contact structures on manifolds are very natural objects, born over two centuries ago, as a natural language for geometric optics, thermodynamics and classical mechanics. Everyday, one encounters contact structures when parallel parking a car, skating, or watching the prismatic play of light in a glass of water. Contact structures have been studied by many mathematicians and seem to touch on diverse areas of mathematics and physics, but only recently have they moved into the foreground of mathematics. This is due to the many remarkable breakthroughs in contact topology, resulting in a rich and beautiful theory with many far reaching applications to areas such as three and four dimensional topology (that is the structure of space and space- time), string theory and large N dualities in modern physics, and fluid dynamics. The Principal Investigator will study the relation between contact geometry and various topological objects (such as open book decompositions and foliations). In addition he will develop analytic and algebraic tools for studying contact structures in high dimensions. As a whole it is expected that at the conclusion of this project we will have a much better understanding of contact structures in all dimensions and see many more surprising implications for low- dimensional manifolds.