Contact Geometry, Contact Homology and Open Book Decompositions

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AbstractAward: DMS-0804820Principal Investigator: John B. EtnyreThe focus of this proposal is to better understand the intriguingnature of contact structures in all (odd) dimensions, withspecial attention given to dimension three and to apply contactgeometric techniques to questions in topology. The first maintheme of the proposed research is open book decompositions. Theconnection of contact geometry and these specially fibered linkshave provided a fount of information concerning contactstructures and low-dimensional topology. As part of this proposalthis fundamental connection will be exploited to define and studynew invariants of contact structures and special knots in them -specifically, Legendrian and transversal knots. From prior workit is clear these invariants have subtle connections to the famedtight vs. overtwisted dichotomy in dimension three and propertiesof symplectic fillings of contact structures, which in turn arecrucial to applications to topology. Some of the major outcomesexpected from this part of the project are various classificationresults for Legendrian and transversal knots in tight andovertwisted contact structures and a better understanding of thefundamental process of contact surgery. The project will alsostudy the connection between contact structures and open books inhigh dimensions with the goal of illuminating existence questionsin high dimensions and trying to generalize the notion ofovertwistedness here as well. Similar to the connection with openbooks, contact structures are also related to foliations. Thisconnection has been instrumental in applications of contactgeometry to low-dimensional topology. Many of the outstandingquestions concerning this connection will be explored. A secondmain theme of the proposal is the development and analysis ofLegendrian contact homology in higher dimensions. There is agreat deal of beautiful analytic and algebraic structure in thistheory, much of which is yet to be discovered. Moreover, throughthe elegant conormal construction one can then use this to defineinvariants of manifolds and of their embeddings in Euclideanspace. These invariants appear to be extremely powerful and willbe thoroughly investigated.Contact structures on manifolds are very natural objects, bornover two centuries ago, as a natural language for geometricoptics, thermodynamics and classical mechanics. Everyday, oneencounters contact structures when parallel parking a car,skating, or watching the prismatic play of light in a glass ofwater. Contact structures have been studied by manymathematicians and seem to touch on diverse areas of mathematicsand physics, but only recently have they moved into theforeground of mathematics. This is due to the many remarkablebreakthroughs in contact topology, resulting in a rich andbeautiful theory with many far reaching applications to areassuch as three and four dimensional topology (that is thestructure of space and space- time), string theory and large Ndualities in modern physics, and fluid dynamics. The PrincipalInvestigator will study the relation between contact geometry andvarious topological objects (such as open book decompositions andfoliations). In addition he will develop analytic and algebraictools for studying contact structures in high dimensions. As awhole it is expected that at the conclusion of this project wewill have a much better understanding of contact structures inall dimensions and see many more surprising implications for low-dimensional manifolds.