Some remarks on the coarse index theorem for complete Riemannian manifolds (Q1429085)

Coarse geometry describes Riemannian manifolds on very large scales. In this theory, coarse cohomology and coarse index theory play the role of singular cohomology and classical index theory; both deal with objects that are invariant under altering the underlying manifolds on small scales. The author computes the coarse index of untwisted Dirac operators in two special cases. Let \(M\) be a compact manifold, let \(N=M\times\mathbb{R}^n\) and let \(D_M\) and \(D_N\) be corresponding untwisted Dirac operators. Then the first result says that the coarse index of \(D_N\) equals \(c_n\operatorname {ind}(D_M)\) for some constant \(c_n>0\). The second result computes the coarse index of \(\mathbb{R}^n\) by relating the total Riesz transformation of \(L^2(\mathbb{R}^n)\) to a generator of the coarse cohomology group \(HX^n(\mathbb{R}^n)\).