Index Theory, Coarse Geometry, and Topology of Manifolds
About this Title
John Roe, University of Oxford, Oxford, England
Publication: CBMS Regional Conference Series in Mathematics
Publication Year 1996: Volume 90
ISBNs: 978-0-8218-0413-1 (print); 978-1-4704-2450-3 (online)
MathSciNet review: MR1399087
MSC: Primary 58G12; Secondary 19J25, 19K35, 19K56, 20F32, 46L80, 57R67
The Atiyah-Singer index theorem is one of the most powerful tools for relating geometry, analysis, and topology. In its original form, however, it applies only to compact manifolds. This book describes a version of index theory which works for noncompact spaces with appropriate control, such as complete Riemannian manifolds. The relevant “control” is provided by the large scale geometry of the space, and basic notions of large scale geometry are developed.
Index theory for the signature operator is related to geometric topology via surgery theory. And, paralleling the analytic development, “controlled” surgery theories for noncompact spaces have been developed by topologists. This book explores the connections between these theories, producing a natural transformation from surgery to “analytic surgery”.
The analytic foundations of the work come from the theory of $C^*$-algebras, and the properties of the $C^*$-algebra of a coarse space are developed in detail.
The book is based on lectures presented at a conference held in Boulder, Colorado, in August 1995 and includes the author's detailed notes and descriptions of some constructions that were finalized after the lectures were delivered.
Also available from the AMS by John Roe is Lectures on Coarse Geometry.
Graduate students and research mathematicians working in global analysis, geometric topology, and infinite group theory.
Table of Contents
- 1. Index theory (Chapter 1)
- 2. Coarse geometry (Chapter 2)
- 3. $C^*$-algebras (Chapter 3)
- 4. An example of a higher index theorem (Chapter 4)
- 5. Assembly (Chapter 5)
- 6. Surgery (Chapter 6)
- 7. Mapping surgery to analysis (Chapter 7)
- 8. The coarse Baum-Connes conjecture (Chapter 8)
- 9. Methods of computation (Chapter 9)
- 10. Coarse structures and boundaries (Chapter 10)