  We will show that for a polynomially contractible manifold of bounded
geometry and of polynomial volume growth every coarse and rough cohomology
class pairs continuously with the K-theory of the uniform Roe algebra. As an
application we will discuss non-vanishing of rough index classes of Dirac
operators over such manifolds, and we will furthermore get higher-codimensional
index obstructions to metrics of positive scalar curvature on closed manifolds
with virtually nilpotent fundamental groups. We will give a computation of the
homology of (a dense, smooth subalgebra of) the uniform Roe algebra of
manifolds of polynomial volume growth.

Differential Geometry;K-Theory and Homology