On the uniformly proper classification of open manifolds

We give a brief account on the uniformly proper classification of open manifolds, i.e., the classification under bounded, uniformly proper maps. The small category of diffeomorphism classes of open $n$-manifolds has uncountably many homotopy types, $n\ge2$. Our main approach consists in splitting this set into generalized components and then try to classify these components and thereafter the elements inside a component. To define these components, we introduce Gromov–Hausdorff and Lipschitz metrizable uniform structures and corresponding $\mathrm{GH}$- and $\mathrm{L}$-cohomologies. The $\mathrm{GH}$-components are particularly appropriate to introduce geometric bordism theory for open manifolds, the $\mathrm{L}$-components are appropriate to establish surgery. We present independent generators for the bordism groups. The fundamental contributions of Farrell, Wagoner, Siebenmann, Maumary, and Taylor play a decisive role. In our approach, we suppose the manifolds to be endowed with a metric of bounded geometry and restrict ourselves to bounded uniformly proper morphisms. Finally, we discuss the question under which conditions bounded geometry and uniform properness are preserved by surgery, and sketch some proper surgery groups.