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

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(2) connections, provided M = S ∪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 [17]. AMS Classification numbers Primary: 57M27 Secondary: 57R57, 53D12, 58J30