The homotopy type of the topological cobordism category (Q2695703)

Summary: We define a cobordism category of topological manifolds and prove that if \(d\neq 4\) 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.