Tight Contact Structures and 3-dimensional Topology

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

Proposal: DMS-0072853 Abstract The investigators propose to explore 3-dimensional topology via contact structures, based on new techniques in the classification of tight contact structures on various 3-dimensional manifolds. Our main goal is to develop the 3-dimensional cut-and-paste techniques involving convex surfaces and ``bypasses" into a largely combinatorial one. The investigators propose to import ideas and constructions from the theory of foliations and laminations (in joint work with J. Etnyre and W. Kazez). Convex surfaces and bypasses aid the decomposition of a tight contact manifold (eventually) into balls, similar to the `` sutured manifold decomposition", due to Gabai. The ``dividing curves on the surfaces along which the cuttings take place determine the tight contact structure. A project which is currently under way is to carefully follow the sutured manifold decompositions of Gabai in constructing taut foliations on most 3-manifolds, and to construct tight contact structures by gluing in much the same way as Gabai's construction. We hope to produce an effective gluing theorem for tight contact structures. Another direction of research is Legendrian knot theory. Using the classification of tight contact structures on solid tori, Etnyre and Honda propose to classify Legendrian torus knots and Legendrian figure eight knots. The investigators propose a study of 3-dimensional spaces. The 3-dimensional spaces we study will locally be similar to the standard Euclidean 3-dimensional space. These objects may be very complicated globally, but a local observer cannot tell the difference, just as an ant cannot tell whether it is sitting on a flat plane or a very large sphere. `Finite' 2-dimensional spaces have been classified and understood for a long time - they are the 2-dimensional sphere, the doughnut, the doughnut with 2 holes, the doughnut with 3 holes, etc., and are distinguished by the number of holes. However, in spite of work by numerous mathematicians this century, a complete classification of 3-dimensional spaces is far from understood. In our work we seek to better understand 3-dimensional spaces by imposing an additional structure, called a contact structure, which, very loosely speaking, amounts to choosing a preferred direction (or a spinning axis) at every point in the 3-dimensional space. Contact structures have intimate connections with 4-dimensional geometry, quantum physics, and dynamics (such as fluid dynamics), and we hope to gain better understanding of 3-dimensional spaces through contact structures.