Holomorphic Disks and Low-Dimensional Topology

Columbia University, New York NY

Investigators

Abstract

DMS-0234311 Peter Ozsvath The proposal deals invariants for three- and four-dimensional manifolds, specifically those which the proposer constructed in collaboration Zoltan Szabo. These invariants appear to be closely related to gauge-theoretic (Donaldson and Seiberg-Witten) invariants, except that they have the advantage of being more combinatorial in nature than their gauge-theoretic predecessors, and hence easier to calculate. To illustrate their strength, one can reprove some results in four-dimensional differential topology which were obtained previously using gauge theory. These invariants also have new topological applications which go beyond gauge theory. For instance, they can be used to give new restrictions on knots whose surgeries give lens spaces. The author hopes to further extend this theory, to see what further insight they may give to topology in dimensions three and four. The proposal deals with studying the intrinsic structure of three- and four-dimensional spaces. The study of four-dimensional spaces was revolutionized by Simon Donaldson in the early 80's by his introduction of "gauge-theoretic techniques". These techniques associate to a space the space of solutions of certain partial differential equations (which come from mathematical physics). Course properties of these solution spaces are then used to give insight into the finer structure of the underlying spaces on which the equations are defined. The proposal deals with efforts at circumventing these (often difficult to identify) auxiliary spaces of solutions, to give more directly combinatorial approaches to understanding the way three- and four-dimensional spaces fit together.