Embedded Cobordism Categories and Spaces of Submanifolds

Galatius, Madsen, Tillmann, and Weiss [7] have identified the homotopy type of the classifying space of the cobordism category with objects (d −1)-dimensional manifolds embedded in ℝ^∞. In this paper we apply the techniques of spaces of manifolds, as developed by the author and Galatius in [8], to identify the homotopy type of the cobordism category with objects (d −1)-dimensional submanifolds of a fixed background manifold M. There is a description in terms of a space of sections of a bundle over M associated to its tangent bundle. This can be interpreted as a form of Poincaré duality, relating a space of submanifolds of M to a space of functions on M.