3-manifolds and Floer homologies

Oklahoma State University, Stillwater OK

Investigators

Abstract

The problems addressed in this project are in the area of Topology. The main theme is to study the fundamental and important properties of the gauge-theoretic/symplectic Floer homology, using invariants and methods from 3-manifold topology and symplectic topology. The investigator studied the semi-infinity of the instanton (gauge theoretic) Floer homology and the intrinsic dependence of the monopole(Seiberg-Witten-Floer) homology of (homology) three-spheres. The major part of the project is to study the interactive relation between the symplectic Floer homology and the classical 3-manifolds, and the relation between the Floer cohomology and the semi-infinite cohomology, and the Seiberg-Witten-Floer theory intertwining the instanton and monopole results. The other part of this project is to study intrinsic properties for the invariants of larger classes of three-spheres.It is a fundamental aim to investigate the change of the new invariants under certain topological operations. The project will integrate the interactions among the instanton theory, the monopole theory, the symplectic Floer theory and the semi-infinite cohomology of infinite-dimensional Lie algebras. A three-dimensional manifold is a space where a nearsighted person sees a standard three-dimensional space everywhere.(Homology) three-spheres are those three-manifolds that one cannot tell from the standard three-sphere by using the usual topological tools. A symplectic manifold is an even-dimensional space with a special(symplectic) structure. For instance, the phase space of a mechanical system is a symplectic space. Such a symplectic structure is rich in mathematics and physics, and is canonical from any nearby region. This shows the subtlety and the complexity of the world we live in. Any local information is no longer useful, the global behavior and the global (topological) invariants are the pivot. The refined invariants studied in this project are intended to distinguish manifolds from the aspects of mathematical physics, and to intertwine various quantum field theories from the aspect of mathematics. Thus it is extremely valuable to investigate the relation between the symplectic Floer cohomology and the semi-infinite cohomology, as well as the interactive link between the instanton homology and the monopole homology. This project addresses some of the most fundamental problems in this subject.