Memoirs of the American Mathematical Society 2012; 92 pp; softcover Volume: 220 ISBN10: 0821872966 ISBN13: 9780821872963 List Price: US$67 Individual Members: US$40.20 Institutional Members: US$53.60 Order Code: MEMO/220/1036
 The authors express the ConnesChern of the Dirac operator associated to a bmetric on a manifold with boundary in terms of a retracted cocycle in relative cyclic cohomology, whose expression depends on a scaling/cutoff parameter. Blowingup the metric one recovers the pair of characteristic currents that represent the corresponding de Rham relative homology class, while the blowdown yields a relative cocycle whose expression involves higher eta cochains and their banalogues. The corresponding pairing formulæ, with relative Ktheory classes, capture information about the boundary and allow to derive geometric consequences. As a byproduct, the authors show that the generalized AtiyahPatodiSinger pairing introduced by Getzler and Wu is necessarily restricted to almost flat bundles. Table of Contents  Introduction
 Preliminaries
 The banalogue of the entire Chern character
 Heat kernel and resolvent estimates
 The main results
 Bibliography
 Subject index
 Notation index
