  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.
