Gauge Theory and Geometry in Dimensions Three and Four

Harvard University, Cambridge MA

Investigators

Abstract

Abstract Award: DMS-0100771. Principal Investigator: Peter B. Kronheimer The aim of this project is to apply gauge-theory techniques to the study of three-dimensional manifolds. The principal investigator proposes to investigate Floer homology and closely related areas of geometry, and hopes to shed light on the applicability of gauge theory to problems in three-dimensional topology. In particular, it is hoped that a relation can be established between the Floer homologies of three-manifolds defined on the one hand by the monopole equations on the other hand by the instanton equations. (These are the equations which, in four-dimensions, lead respectively to the Seiberg-Witten invariants and Donaldson invariants of four-manifolds, and which have led to an flood of results in four-dimensional differential topology in the past twenty years.) A first goal is to prove that if the instanton Floer homology of manifold with first betti number one is trivial in the strong sense that all the representations of the fundamental group can be made to disappear by a holonomy perturbation, then the monopole Floer homology groups are trivial also. (For the instanton groups, the relevant representations are the representations in SO(3) with non-trivial Stiefel-Whitney class.) By an application of a non-vanishing theorem for the monopole Floer homology and use of Floer's exact triangle, this would lead to a proof of the "Property P conjecture". A related goal in this project is the development of new constructions for Floer homology, based on the technique of finite-dimensional approximation (which has already seen convincing application in the study of the four-dimensional invariants). Topology is the qualitative study of space and its connectedness. Its importance was recognized at the turn of the last century by the French mathematician Poincaro, during his investigation of the laws of motion that govern the movement of a three-body system such as the Earth, Moon and Sun moving according to Newton's laws. In the past twenty years, topology has seen applications in questions such as the knotting of proteins and DNA, and in modern theories of high-energy physics. The topology of three-dimensional spaces, as opposed to those of higher dimension, is of particular subtlety. Through this project, it is hoped to bring new techniques to bear on outstanding questions in three-dimensional topology. These techniques -- gauge theory and the Seiberg-Witten equations -- originated in physics, where they had potential application to fundamental questions such as quark confinement. They have been an effective tool in the study of four-dimensional spaces (such as our space-time). The aim now is to apply the same techniques to questions in dimension three.