On the rho invariant for manifolds with boundary

Abstract

This article is a follow up of the previous article of the authors on the analytic surgery of $\eta$– and $\rho$–invariants. We investigate in detail the (Atiyah–Patodi–Singer) $\rho$–invariant for manifolds with boundary. First we generalize the cut-and-paste formula to arbitrary boundary conditions. A priori the $\rho$–invariant is an invariant of the Riemannian structure and a representation of the fundamental group. We show, however, that the dependence on the metric is only very mild: it is independent of the metric in the interior and the dependence on the metric on the boundary is only up to its pseudo–isotopy class. Furthermore, we show that this cannot be improved: we give explicit examples and a theoretical argument that different metrics on the boundary in general give rise to different $\rho$–invariants. Theoretically, this follows from an interpretation of the exponentiated $\rho$–invariant as a covariantly constant section of a determinant bundle over a certain moduli space of flat connections and Riemannian metrics on the boundary. Finally we extend to manifolds with boundary the results of Farber–Levine–Weinberger concerning the homotopy invariance of the $\rho$–invariant and spectral flow of the odd signature operator.