Cobordism categories, corners, and surgery

The first result of this manuscript is a formula for calculating the homotopy type of the classifying space of the cobordism category Cobqd,k the cobordism category whose morphisms are cobordisms of manifolds of fixed dimension d with corners of codimension ≤ k together with structure on the tangent bundle determined by a fibration theta. The result is the zero space of a homotopy colimit over a certain diagram of Thom spectra. In some interesting cases we are able to evaluate this homotopy colimit explicitly. One such case is the cobordism category of oriented two-dimensional manifolds with boundary. In other words, we determine the homotopy type of the domain of an open-closed topological field theory.;The second half of this thesis shows how the traditional sets appearing in the Browder-Novikov-Sullivan-Wall smooth surgery sequence may be replaced by the classifying spaces of cobordism categories and the maps between these sets may be replaced by functors.