Slowly Increasing Cohomology for Discrete Metric Spaces with Polynomial Growth

The authors introduce a kind of slowly increasing cohomology HS* (X) for a discrete metric space X with polynomial growth, and construct a character map from the slowly increasing cohomology HS* (X) into HC*_cont(S(X)), the continuous cyclic cohomology of the smooth subalgebra S(X) of the uniform Roe algebra B*(X). As an application, it is shown that the fundamental cocycle, associated with a uniformly contractible complete Riemannian manifold M with polynomial volume growth and polynomial contractibility radius growth, is slowly increasing.