Indices of pseudodifferential operators on open manifolds

Abstract

We generalize Roe's Index Theorem for operators of Dirac type on open manifolds to elliptic pseudodifferential operators. To this end we introduce a class of pseudodifferential operators on manifolds of bounded geometry which is more general than similar classes defined by other authors. We revisit Spakula's uniform K-homology and show that our elliptic pseudodifferential operators naturally define classes there. Furthermore, we use the uniform coarse assembly map to relate these classes to the index classes of these operators in the K-theory of the uniform Roe algebra. Our investigation of uniform K-homology goes on with constructing the external product for it and deducing homotopy invariance. The next major result is the identification of the dual theory of uniform K-homology: uniform K-theory. We give a simple definition of uniform K-theory for all metric spaces and in the case of manifolds of bounded geometry we give an interpretation via vector bundles of bounded geometry over the manifold. This opens up the door for Chern-Weil theory and we define a Chern character map from uniform K-theory of a manifold to its bounded de Rham cohomology. We introduce a type of Mayer-Vietoris argument for these uniform (co-)homology theories which enables us to show firstly, that the Chern character induces an isomorphism modulo torsion, and secondly, that we have Poincare duality between uniform K-theory and uniform K-homology if the manifold is spin-c. Poincare duality together with the relation of uniform K-homology to the index theorem of Roe mentioned above directly leads to a generalization of the index theorem to elliptic pseudodiffential operators. Finally, using homotopy invariance of uniform K-homology we derive important results about the uniform coarse Baum-Connes conjecture establishing it equally important as the usual coarse Baum-Connes conjecture.