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On the rho invariant for manifolds with boundary

Paul A Kirk and Matthias Lesch

Algebraic & Geometric Topology 3 (2003) 623–675

arXiv: math.DG/0203097


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

$\rho$–invariant, $\eta$–invariant
Mathematical Subject Classification 2000
Primary: 58J28
Secondary: 57M27, 58J32, 58J30
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Received: 30 January 2003
Accepted: 4 June 2003
Published: 25 June 2003
Paul A Kirk
Department of Mathematic
Indiana University
Bloomington, IN 47405
Matthias Lesch
Universität zu Köln
Mathematisches Institut
Weyertal 86–90
50931 Köln