Pseudoholomorphic curves in low-dimensional topology

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

The goal of this project is to develop new topological invariants in the framework of Floer homology. A major part of the project is to develop "embedded contact homology", a new invariant of contact three-manifolds which counts embedded pseudoholomorphic curves in the four-dimensional symplectization of the three-manifold. Embedded contact homology is conjecturally isomorphic to a version of the Seiberg-Witten or Ozsvath-Szabo Floer homologies. It provides a bridge between the topology of smooth manifolds in three and four dimensions, and the geometry and dynamics of holomorphic curves and contact structures. Another part of the project is to construct Floer-theoretic invariants of families of equivalent objects for different versions of Floer theory, thus obtaining homotopy invariants of families of three-manifolds, symplectomorphisms, Legendrian knots, and other types of objects for which Floer theory can be defined. This project fits into the broad theme of developing tools to understand the possible global shapes of three- and four-dimensional spaces, such as the universe that we live in. The tools used here to understand the shape of a space involve counting interesting geometric objects inside the space. An important class of such objects are pseudoholomorphic curves, which are surfaces resembling soap films. By counting the number of such surfaces that exist in a space, one can gain deep information about the global structure of the space.