Some Remarks on the Coarse Index Theorem for Complete Riemannian Manifolds

Abstract

In this paper, we state two results on the coarse index for simple models of the coarse geometry. The first one is a coarse index theorem on trivial bundles over even dimensional closed Riemannian manifolds. It comes from the Poincar\'e duals for these base manifold in the trivial bundles. We next show some correspondence between the generator of the coarse cohomology of $\mathbf{R}^n$ and the Riesz transformation on $\mathbf{R}^n$.