Coarse Cohomology and Index Theory on Complete Riemannian Manifolds

"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.

''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.

We develop a link between the 'coarse' geometry of complete Riemannian manifolds and index theory for elliptic operators on such manifolds. We define and make use of a new cohomology theory that is sensitive only to this coarse geometry. The connection with index theory is made by a character map between this coarse cohomology theory and the cyclic cohomology of an operator algebra whose [italic capital]K-theory is the receptacle for an abstract index