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.

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Paul Kirk, Matthias Lesch
Department of Mathematics, Indiana University
Bloomington, IN 47405, USA
Universitat zu Koln, Mathematisches Institut
Weyertal 86-90, 50931 Koln, Germany


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