Singular Connections on Three-Manifolds and Manifolds with Cylindrical Ends

Abstract

This thesis studies moduli spaces of singular connections on 3-manifolds and manifolds with cylindrical ends. A Chern-Simons functional is defined for singular connections on 3-manifolds which are singular along a knot. The critical points of that Chern-Simons functional are flat singular connections. The Hodge-de Rham theory of singular flat connections is presented, yielding a topological condition for determining when a singular flat connection is isolated. Finally, moduli spaces of anti-self-dual singular connections are investigated on manifolds with cylindrical ends. The main analytical tool used to study singular connections is the theory of elliptic edge operators, summarized in the appendix. It is hoped that this work can lead to defining a Floer homology for singular connections.