A splitting formula for spectral flow on closed 3-manifolds

In 1989 Edward Witten defined a topological quantum field theory using certain 3-manifold invariants involving the Feynman path integral. This integral is not mathematically rigorous, but Witten's invariants have two different interpretations. A comparison of these predicts a relationship between an entirely combinatorial description in terms of the axioms of topological quantum field theory and gauge theoretic quantities. In 1992 Lisa Jeffrey verified that the TQFT and the asymptotic expansion are consistent for lens spaces. She also confirmed this for torus bundles over the circle but left some details involving spectral flow unresolved. Based on her analysis she made a conjecture involving the spectral flow of the odd signature operator between flat SU(2) connections on torus bundles over the circle.; This thesis establishes a splitting formula for spectral flow of the odd signature operator between flat SU(2) connections on a closed 3-manifold M. It describes spectral flow on M = S ∪ X in terms of spectral flow on the solid torus S, spectral flow on its complement X (with certain Atiyah-Patodi-Singer boundary conditions), and two correction terms which depend only on the endpoints. The central ingredient to the proof of this theorem is a result by Liviu Nicolaescu about the relationship of spectral flow and the Maslov index of Cauchy data spaces, as well as extensions by Hans Boden, Chris Herald, Mark Daniel, Paul Kirk, Erik Klassen and Matthias Lesch.; The main application of this splitting formula is the proof of the above conjecture by Lisa Jeffrey. A major part of this dissertation lies in the technical issues involving the Atiyah-Patodi-Singer boundary conditions and in the detailed analysis of spectral flow on S with these boundary conditions. This is used in this thesis for the study of spectral flow on torus bundles M over the circle, which also relies on computations of the twisted cohomology of X = M - S, where S is a certain embedded solid torus.