Abstract
For a closed, oriented, odd dimensional manifold X, we define the rho invariant rho(X,E,H) for the twisted odd signature operator valued in a flat hermitian vector bundle E, where H = \sum i^{j+1} H_{2j+1} is an odd-degree closed differential form on X and H_{2j+1} is a real-valued differential form of degree {2j+1}. We show that the twisted rho invariant rho(X,E,H) is independent of the choice of metrics on X and E and of the representative H in the cohomology class [H]. We establish some basic functorial properties of the twisted rho invariant. We express the twisted eta invariant in terms of spectral flow and the usual eta invariant. In particular, we get a simple expression for it on closed oriented 3-dimensional manifolds with a degree three flux form. A core technique used is our analogue of the Atiyah-Patodi-Singer theorem, which we establish for the twisted signature operator on a compact, oriented manifold with boundary. The homotopy invariance of the rho invariant rho(X,E,H) is more delicate to establish, and is settled under further hypotheses on the fundamental group of X.