Global Analysis on Foliated Spaces

This book presents a complete proof of Connes' Index Theorem generalized to foliated spaces, including coverage of new developments and applications.

Calvin C. Moore (Author), Claude L. Schochet (Author)

9780521613057, Cambridge University Press

Paperback, published 19 December 2005

308 pages
23.6 x 15.6 x 1.8 cm, 0.44 kg

'This book presents a complete proof of this beautiful result, generalized to foliated spaces (not just manifolds). It includes the necessary background from analysis geometry and topology. This second edition has improved exposition, an updated bibliography, an index, and additional material covering developments and applications since the first edition came out.' L'enseignement mathematique

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 and topology. 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 second edition presents a complete proof of this beautiful result, generalized to foliated spaces (not just manifolds). It includes the necessary background from analysis, geometry, and topology. The present edition has improved exposition, an updated bibliography, an index, and additional material covering developments and applications since the first edition came out, including the confirmation of the Gap Labeling Conjecture of Jean Bellissard.

Introduction
1. Locally traceable operators
2. Foliated spaces
3. Tangential cohomology
4. Transverse measures
5. Characteristic classes
6. Operator algebra
7. Pseudodifferential operators
8. The index theorem
Appendices.

Subject Areas: Topology [PBP], Geometry [PBM], Complex analysis, complex variables [PBKD], Real analysis, real variables [PBKB]