Algebraic and Geometric Topology 3 (2003),
paper no. 22, pages 623-675.
On the rho invariant for manifolds with boundary
Paul Kirk, Matthias Lesch
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.
Keywords.
rho-invariant, eta-invariant
AMS subject classification.
Primary: 58J28.
Secondary: 57M27, 58J32, 58J30.
DOI: 10.2140/agt.2003.3.623
E-print: arXiv:math.DG/0203097
Submitted: 30 January 2003.
Accepted: 4 June 2003.
Published: 25 June 2003.
Notes on file formats
Paul Kirk, Matthias Lesch
Department of Mathematics, Indiana University
Bloomington, IN 47405, USA
and
Universitat zu Koln, Mathematisches Institut
Weyertal 86-90, 50931 Koln, Germany
Email: pkirk@indiana.edu, lesch@mi.uni-koeln.de
URL:
http://php.indiana.edu/~pkirk,
http://www.mi.uni-koeln.de/~lesch
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