Gauge Theory and Geometry in Dimensions Three and Four

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AbstractAward: DMS-0100771.Principal Investigator: Peter B. KronheimerThe aim of this project is to apply gauge-theory techniques tothe study of three-dimensional manifolds. The principalinvestigator proposes to investigate Floer homology and closelyrelated areas of geometry, and hopes to shed light on theapplicability of gauge theory to problems in three-dimensionaltopology. In particular, it is hoped that a relation can beestablished between the Floer homologies of three-manifoldsdefined on the one hand by the monopole equations on the otherhand by the instanton equations. (These are the equations which,in four-dimensions, lead respectively to the Seiberg-Witteninvariants and Donaldson invariants of four-manifolds, and whichhave led to an flood of results in four-dimensional differentialtopology in the past twenty years.) A first goal is to provethat if the instanton Floer homology of manifold with first bettinumber one is trivial in the strong sense that all therepresentations of the fundamental group can be made to disappearby a holonomy perturbation, then the monopole Floer homologygroups are trivial also. (For the instanton groups, the relevantrepresentations are the representations in SO(3) with non-trivialStiefel-Whitney class.) By an application of a non-vanishingtheorem for the monopole Floer homology and use of Floer's exacttriangle, this would lead to a proof of the "Property Pconjecture". A related goal in this project is the developmentof new constructions for Floer homology, based on the techniqueof finite-dimensional approximation (which has already seenconvincing application in the study of the four-dimensionalinvariants).Topology is the qualitative study of space and its connectedness.Its importance was recognized at the turn of the last century bythe French mathematician Poincaro, during his investigation ofthe laws of motion that govern the movement of a three-bodysystem such as the Earth, Moon and Sun moving according toNewton's laws. In the past twenty years, topology has seenapplications in questions such as the knotting of proteins andDNA, and in modern theories of high-energy physics. The topologyof three-dimensional spaces, as opposed to those of higherdimension, is of particular subtlety. Through this project, itis hoped to bring new techniques to bear on outstanding questionsin three-dimensional topology. These techniques -- gauge theoryand the Seiberg-Witten equations -- originated in physics, wherethey had potential application to fundamental questions such asquark confinement. They have been an effective tool in the studyof four-dimensional spaces (such as our space-time). The aim nowis to apply the same techniques to questions in dimension three.