The homotopy type of the topological cobordism category

Abstract

We define a cobordism category of topological manifolds and prove that if $d\ne 4$ its classifying space is weakly equivalent to ${\Omega }^{\infty -1}MT\mathrm{Top}(d)$, where $MT\mathrm{Top}(d)$ is the Thom spectrum of the inverse of the canonical bundle over $B\mathrm{Top}(d)$. We also give versions for manifolds with tangential structures and/or boundary. The proof uses smoothing theory and excision in the tangential structure to reduce the statement to the computation of the homotopy type of smooth cobordism categories due to Galatius–Madsen–Tillman–Weiss.