  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 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.
