Cyclic Cohomology and Higher Indexes for Noncompact Complete Manifolds

The indices of generalized Dirac operators on noncompact complete Riemannian manifolds live in the K theory of (uniform) Roe algebras. In this paper we shall compute the cyclic cohomology of (uniform) Roe algebras associated to noncompact manifolds. The cyclic cohomology of the (uniform) Roe algebras is identified with (uniform) simplical cohomology with infinite support of the Rips′ polyhedrons associated to a net of the Riemannian manifold. We also compute the Chern character of the K theoretic indices of generalized Dirac operators. We apply such a computation to the analysis and geometry of noncompact complete Riemannian manifolds. In particular we show that a uniformly contractible Riemnannian manifold with bounded geometry cannot have uniform positive scalar curvature outside a compact set if the volume and contractibility radius have certain subexponential growth.