A Classification Approach for Open Manifolds

There are a least uncountably many diffeomorphism types for open manifolds. Hence the classification problem is extremely difficult. We proceed as follows: We define several uniform structures of proper metric spaces and consider their arc components. Any open complete manifold (M^n,g) defines such a component. Hence the classification amounts to two steps. First counting all components, and secondly counting all elements inside a component. This can be partially done by invariants. We define bordism groups and relative characteristic numbers.