A classification approach for open manifolds

On the set of all smooth Riemannian manifolds, a number of different uniform structures are studied, some of which are defined for the more general class of proper metric spaces, while the others take into account the smooth and Riemannian structures. Appropriate bordism theories are presented. Several homology theories that do not distinguish manifolds in one component of a given uniform structure, but can distinguish manifolds in different components are discussed. The proofs are omitted.