  This paper describes a topological method to compute the spectral flow of a
family of twisted Dirac operators, it includes two detailed examples. Briefly,
a formula of Atiyah, Patodi and Singer expresses the spectral flow in terms of
Chern-Simons invariants and rho invariants. The first step is to construct a
flat cobordism to a new bigger 3-manifold. The advantage of the new connection
is that it is in the path component of a reducible connection. The second step
is to calculate the effect of these operations on the invariants. The final
step is an application of the G-signature theorem to compute the invariants.
