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＊ont （S（X））, the continuous cyclic cohomol- ogy 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.