# Connes-Chern Character for Manifolds with Boundary and Eta Cochains

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*Matthias Lesch; Henri Moscovici; Markus J. Pflaum*

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

# Table of Contents

## Connes-Chern Character for Manifolds with Boundary and Eta Cochains

- List of Figures vii8 free
- Introduction 110 free
- Chapter 1. Preliminaries 918 free
- 1.1. The general setup 918
- 1.2. Relative cyclic cohomology 1019
- 1.3. The Chern character 1221
- 1.4. Dirac operators and 𝑞-graded Clifford modules 1322
- 1.5. The relative Connes–Chern character of a Dirac operator over a manifold with boundary 1524
- 1.6. Exact b-metrics and b-functions on cylinders 1726
- 1.7. Global symbol calculus for pseudodifferential operators 1928
- 1.8. Classical b-pseudodifferential operators 2029
- 1.9. Indicial family 2433

- Chapter 2. The b-Analogue of the Entire Chern Character 2534
- 2.1. The b-trace 2534
- 2.2. The relative McKean–Singer formula and the APS Index Theorem 2837
- 2.3. A formula for the b-trace 3039
- 2.4. b-Clifford modules and b-Dirac operators 3342
- 2.5. The b-JLO cochain 3544
- 2.6. Cocycle and transgression formulæ for the even/odd b-Chern character (without Clifford covariance) 3645
- 2.7. Sketch of Proof of Theorem 2.11 3948

- Chapter 3. Heat Kernel and Resolvent Estimates 4554
- 3.1. Basic resolvent and heat kernel estimates on general manifolds 4554
- 3.2. Comparison results 5261
- 3.3. Trace class estimates for the model heat kernel 5564
- 3.4. Trace class estimates for the JLO integrand on manifolds with cylindrical ends 5867
- 3.5. Estimates for b-traces 5968
- 3.6. Estimates for the components of the entire b-Chern character 6069

- Chapter 4. The Main Results 6574
- Bibliography 8594
- Subject Index 8998
- Notation Index 91100