The pioneering works of Bennequin and Eliashberg from the 1980s and the 1990s have shown that, among all contact structures, the tight ones best reflect the topology of the ambient space. Only in the past decade have sufficient tools been developed to allow many of the basic classification questions in the subject to be attacked. The PI proposes joint work with Honda and Mati'c to analyze the contact topology of Haken manifolds using cut-and-paste techniques. Important outgrowths of this investigation include constructions of tight contact structures in Haken homology spheres, extensions of the relationship with foliation theory to laminations and hyperbolic manifolds, and an understanding of the interplay between the subsurface complex and contact structures on a product.
Three-dimensional manifolds are modeled on the three-dimensional space we live in. Contact and symplectic topology were originally motivated by questions of physicists working in classical mechanics and thermodynamics. They arise in a surprisingly broad array of applied situations ranging from fluid flows to the manufacture of projection screen televisions. The proposed research has the potential to produce new results applicable to these fields.
