Pseudoholomorphic Curves in Low-Dimensional Topology

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

DMS-0204681 Michael Hutchings This project involves the development and application of new enumerative invariants in low-dimensional, symplectic, and contact topology. There are two main goals. The first is to develop and compute ``periodic Floer homology'', a theory defined for an area-preserving surface diffeomorphism, which counts periodic orbits together with embedded pseudoholomorphic curves in R cross the mapping torus. This theory is conjectured to agree with the Seiberg-Witten Floer homology of the mapping torus, thus giving a link between low dimensional topology and surface dynamics. Also, an analogue of periodic Floer homology for three-dimensional contact manifolds should have applications to the topology of overtwisted contact structures and smooth four-manifolds. The second main goal is to develop and compute invariants of families of equivalent objects in different versions of Floer theory. These give topological invariants of families of symplectomorphisms, three-manifolds, Legendrian knots, and any other type of object for which a version of 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. For example the universe we live in is a four dimensional space if one includes time, and its global structure is not known. 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 with appropriate constraints that exist in a space, one can gain information about the global structure of the space.