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Connes-Chern Character for Manifolds with Boundary and Eta Cochains
Matthias Lesch, Universität Bonn, Germany, Henri Moscovici, Ohio State University, Columbus, OH, and Markus J. Pflaum, University of Colorado, Boulder, CO

Memoirs of the American Mathematical Society
2012; 92 pp; softcover
Volume: 220
ISBN-10: 0-8218-7296-6
ISBN-13: 978-0-8218-7296-3
List Price: US$67
Individual Members: US$40.20
Institutional Members: US$53.60
Order Code: MEMO/220/1036
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The authors 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 formulæ, with relative K-theory classes, capture information about the boundary and allow to derive geometric consequences. As a by-product, the authors show that the generalized Atiyah-Patodi-Singer pairing introduced by Getzler and Wu is necessarily restricted to almost flat bundles.

Table of Contents

  • Introduction
  • Preliminaries
  • The b-analogue of the entire Chern character
  • Heat kernel and resolvent estimates
  • The main results
  • Bibliography
  • Subject index
  • Notation index
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