Contact structures, Floer homology and TQFT

University Of Southern California, Los Angeles CA

Investigators

Abstract

Abstract Award: DMS-0805352 Principal Investigator: Ko Honda Recently it has become more evident that contact structures play a central role in low-dimensional topology and Topological Quantum Field Theories (TQFTs). The cut-and-paste theory of contact structures in dimension three, developed by numerous authors including the PI, can now be imported into Heegaard Floer homology, a (3+1)-dimensional TQFT. The PI, together with his collaborators, seeks to better understand the relations among contact geometry, Floer homology theories, TQFTs and string theory, and hyperbolic geometry. More specifically, the PI proposes the following: (1) Analyze the TQFT properties of Heegaard/sutured Floer homology. (2) Understand a certain category obtained via contact geometry, which satisfies many of the properties of a triangulated category. (3) Study contact homology invariants of exact Lagrangian cobordisms between Legendrian links in the symplectization of a contact manifold. (4) Calculate the contact homology of tight contact structures on hyperbolic 3-manifolds. The PI proposes a study of 3- and 4-dimensional spaces using a probe called a contact structure. 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. A contact structure is a field in 3-dimensional space, analogous to an electric field in physics. Just as the nature of the electric field, such as the strength and the direction at any point, gives some information about the distribution of the electric charge, i.e., the source of the electric field, a contact structure gives information about the shape of the ambient 3-dimensional space. A more thorough mathematical understanding of contact structures in dimension three may contribute significantly towards our understanding of the shape of the universe.