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# memo_has_moved_text(); Connes–Chern character for manifolds with boundary and eta cochains

Matthias Lesch, Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany, Henri Moscovici, Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, USA and Markus J. Pflaum, Department of Mathematics, University of Colorado UCB 395, Boulder, Colorado 80309

Publication: Memoirs of the American Mathematical Society
Publication Year 2012: Volume 220, Number 1036
ISBNs: 978-0-8218-7296-3 (print); 978-0-8218-9209-1 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2012-00656-3
Published electronically: March 13, 2012
Keywords: Manifolds with boundary, b-calculus, noncommutative geometry, Connes–Chern character, relative cyclic cohomology, $\eta$-invariant
MSC (2010): Primary 58Jxx, 46L80; Secondary 58B34, 46L87

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Chapters

• Introduction
• Chapter 1. Preliminaries
• Chapter 2. The $b$-analogue of the entire Chern character
• Chapter 3. Heat kernel and resolvent estimates
• Chapter 4. The main results

### Abstract

We express the Connes-Chern of the Dirac operator associated to a b-metric on a manifold with boundary in terms of a retracted cocycle in relative cyclic cohomology, whose expression depends on a scaling/cut-off parameter. Blowing-up the metric one recovers the pair of characteristic currents that represent the corresponding de Rham relative homology class, while the blow-down yields a relative cocycle whose expression involves higher eta cochains and their b-analogues. The corresponding pairing formulæwith relative K-theory classes capture information about the boundary and allow to derive geometric consequences. As a by-product, we show that the generalized Atiyah-Patodi-Singer pairing introduced by Getzler and Wu is necessarily restricted to almost flat bundles.