This article is a follow up of the previous article of the authors on the analytic surgery of
– and
–invariants.
We investigate in detail the (Atiyah–Patodi–Singer)
–invariant
for manifolds with boundary. First we generalize the cut-and-paste
formula to arbitrary boundary conditions. A priori the
–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
–invariants.
Theoretically, this follows from an interpretation of the exponentiated
–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
–invariant
and spectral flow of the odd signature operator.
