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 If-theory is the receptacle for an abstract index.
K E Y W O R DS AND PHRASES: Course geometry, cyclic cohomology, Atiyah-Singer index theo-
rem, X-theory for C*-algebras, ideal boundary, Novikov conjecture.