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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
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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

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.