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).
Author(s): Calvin C. Moore, Claude L. Schochet
Series: Mathematical Sciences Research Institute Publications
Edition: 2
Publisher: Cambridge University Press
Year: 2005
Language: English
Pages: 308
Cover......Page 1
Half-title......Page 5
Series-title......Page 6
Title......Page 7
Copyright......Page 8
Dedication......Page 9
Contents......Page 11
Preface to the Second Edition......Page 13
Preface to the First Edition......Page 15
Introduction......Page 17
II. Foliated Spaces......Page 24
IV. Transverse Measures......Page 25
VII. Pseudodifferential Operators......Page 26
VIII. The Index Theorem......Page 27
Appendices......Page 28
CHAPTER I Locally Traceable Operators......Page 29
Update 2004......Page 44
CHAPTER II Foliated Spaces......Page 47
Update 2004......Page 70
CHAPTER III Tangential Cohomology......Page 71
Appendix......Page 87
Update 2004......Page 89
CHAPTER IV Transverse Measures......Page 91
Update 2004......Page 124
CHAPTER V Characteristic Classes......Page 125
Update 2004......Page 143
CHAPTER VI Operator Algebras......Page 145
Update 2004......Page 182
CHAPTER VII Pseudodifferential Operators......Page 183
VII-A. Pseudodifferential Operators......Page 185
VII-B. Differential Operators and Finite Propagation......Page 205
VII-C. Dirac Operators and the McKean–Singer Formula......Page 213
VII-D. Superoperators and the Asymptotic Expansion......Page 218
Update 2004......Page 224
CHAPTER VIII The Index Theorem......Page 225
Update 2004......Page 239
CONTENTS......Page 241
A1. Average Euler Characteristic......Page 242
A2. The…-Index Theorem and Riemann–Roch......Page 245
A3. Foliations by Surfaces (Complex Lines or k = 1)......Page 247
A4. Geometric K-Theories......Page 252
A5. Examples of Complex Foliations of Three-Manifolds......Page 258
Update 2004......Page 264
APPENDIX B L2 Harmonic Forms on Noncompact Manifolds......Page 265
APPENDIX C Positive Scalar Curvature Along the Leaves......Page 271
Featured Review, Mathematical Reviews, 2005f:46121......Page 275
References......Page 283
Notation......Page 296
Index......Page 301
