Computing secondary and spectral invariants.

Abstract

In this dissertation we compute invariants which are fundamental to the application of gauge theory in three dimensions; Chern-Simons invariants and rho invariants. We compute these invariants for 3-manifolds with some type of fiber structure. In particular, we compute the Chern-Simons invariants of the binary-dihedral representations which are abelian on the fiber of 3-manifolds which fiber over the circle. Additionally, we compute the Chern-Simons and rho invariants of all representations on each Seifert fiber space. We obtain the Chern-Simons results using three different techniques. The Chern-Simons invariants of the representations on spaces which fiber over a circle are computed directly using Stokes' theorem. In order to compute the Chern-Simons invariants of representations on Seifert fiber spaces, we use a cut-and-paste formula along tori combined with a cobordism argument. To compute the rho invariants, we use two theorems of Atiyah, Patodi and Singer in conjunction with our cobordism argument.