A splitting formula for the spectral flow of the odd signature operator on 3–manifolds coupled to a path of $SU(2)$ connections

Abstract

We establish a splitting formula for the spectral flow of the odd signature operator on a closed $3$–manifold $M$ coupled to a path of $SU\left(2\right)$ connections, provided $M=S\cup X$, where $S$ is the solid torus. It describes the spectral flow on $M$ in terms of the spectral flow on $S$, the spectral flow on $X$ (with certain Atiyah–Patodi–Singer boundary conditions), and two correction terms which depend only on the endpoints.

Our result improves on other splitting theorems by removing assumptions on the non-resonance level of the odd signature operator or the dimension of the kernel of the tangential operator, and allows progress towards a conjecture by Lisa Jeffrey in her work on Witten’s $3$–manifold invariants in the context of the asymptotic expansion conjecture.