Global Analysis on Foliated Spaces

by Calvin C. Moore , Claude L. Schochet

• ISBN13: 9780521613057
• ISBN10: 0521613051
• Edition: 1st
• Format: Paperback
• Copyright: 2005-12-19
• Publisher: Cambridge University Press

Summary

Foliated spaces look locally like products, but their global structure is generally not a product, and tangential differential operators are correspondingly more complex. In the 1980s, Alain Connes founded what is now known as noncommutative geometry. One of the first results was his generalization of the Atiyah-Singer index theorem to compute the analytic index associated with a tangential (pseudo)-differential operator and an invariant transverse measure on a foliated manifold, in terms of topological data on the manifold and the operator. This book presents a complete proof of this beautiful result, generalized to foliated spaces (not just manifolds).

Table of Contents

Preface to the Second Edition	xi
Preface to the First Edition	xiii
Introduction	1	(12)
Locally Traceable Operators	13	(18)
Foliated Spaces	31	(24)
Tangential Cohomology	55	(20)
Transverse Measures	75	(34)
Characteristic Classes	109	(20)
Operator Algebras	129	(38)
Pseudodifferential Operators	167	(42)
Pseudodifferential Operators	169	(21)
Differential Operators and Finite Propagation	190	(7)
Dirac Operators and the McKean--Singer Formula	197	(5)
Superoperators and the Asymptotic Expansion	202	(7)
The Index Theorem	209	(16)
Appendix A. The ∂-Operator	225	(24)
Steven Hurder
Average Euler Characteristic	226	(3)
The ∂-Index Theorem and Riemann-Roch	229	(2)
Foliations by Surfaces (Complex Lines or k = 1)	231	(5)
Geometric K-Theories	236	(6)
Examples of Complex Foliations of Three-Manifolds	242	(7)
Appendix B. L2 Harmonic Forms on Noncompact Manifolds	249	(6)
Calvin C. Moore
Claude Schochet
Robert J. Zimmer
Appendix C. Positive Scalar Curvature Along the Leaves	255	(4)
Robert J. Zimmer
Appendix D. Gap Labeling	259	(8)
Claude Schochet
References	267	(13)
Notation	280	(5)
Index	285