THE HOMOTOPY TYPE OF THE TOPOLOGICAL COBORDISM CATEGORY

We define a cobordism category of topological manifolds and prove that if d not equal 4 its classifying space is weakly equivalent to Omega(infinity-1)MTTop(d), where MTTop(d) is the Thom spectrum of the inverse of the canonical bundle over BTop(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.