Coarse cohomology and index theory on complete Riemannian manifolds
About this Title
Publication: Memoirs of the American Mathematical Society
Publication Year 1993: Volume 104, Number 497
ISBNs: 978-0-8218-2559-4 (print); 978-1-4704-0074-3 (online)
MathSciNet review: 1147350
MSC (1991): Primary 58G12; Secondary 19K56, 55N35
“Coarse geometry” is the study of metric spaces from the asymptotic point of view: two metric spaces (such as the integers and the real numbers) which “look the same from a great distance” are considered to be equivalent. This book develops a cohomology theory appropriate to coarse geometry. The theory is then used to construct “higher indices” for elliptic operators on noncompact complete Riemannian manifolds. Such an elliptic operator has an index in the $K$-theory of a certain operator algebra naturally associated to the coarse structure, and this $K$-theory then pairs with the coarse cohomology. The higher indices can be calculated in topological terms thanks to the work of Connes and Moscovici. They can also be interpreted in terms of the $K$-homology of an ideal boundary naturally associated to the coarse structure. Applications to geometry are given, and the book concludes with a discussion of the coarse analog of the Novikov conjecture.
Researchers in global analysis and geometry.
Table of Contents
- 1. Introduction
- 2. Basic properties of coarse cohomology
- 3. Computation of coarse cohomology
- 4. Cyclic cohomology and index theory
- 5. Coarse cohomology and compactification
- 6. Examples and applications