An approach to classification of open manifolds (Q1404793)

For closed manifolds, there are many approaches to classify them, the Pontryagin-Thorn construction, index theory, bordism theory, surgery, Morse theory, Wall groups, intersection theory, Browder-Novikov-Sullivan-Wall exact theory, \(K\)-theory, etc. However, these approaches are not effective for open manifolds. In this paper, the author investigates various uniform structures on the set of smooth Riemannian manifolds, uniform structures of proper metric spaces, nonlinear analysis on Sobolev spaces on open manifolds, uniform structures of open manifolds, the bordism group for open manifolds and invariants of open manifolds, i.e. \(K\)-theory, rough homology, uniformly locally finite homology, reduced simplicial \(L_{p}\)-cohomology: the author defines uniform structures on the Riemannian manifolds \((M, g)\) which are considered as metric spaces, and decompose the set of all pairs \((M, g)\) into components of a certain uniform structures. He defines two classes of invariants, one for components and one for the manifolds inside a component and then classifies the manifolds in the component under consideration. Moreover, he considers several uniform structures, which become finer and finer. This implies the components become smaller and smaller. In this paper, the author presents the main steps and sketches the proofs.