The homotopy type of the PL cobordism category. II

In this article, we prove the PL analogue of the theorem of Galatius, Madsen, Tillmann, and Weiss which describes the homotopy type of the smooth cobordism category. More specifically, we introduce the PL Madsen-Tillmann spectrum $\mathbf{M}\mathbf{T}PL(d)$ and prove that there is a weak homotopy equivalence of the form $B{\mathsf{C}\mathsf{o}\mathsf{b}}_d^{\mathrm{P}\mathrm{L}}\simeq {\mathrm{\Omega }}^{\mathrm{\infty }-1}\mathbf{M}\mathbf{T}PL(d)$. We also discuss how to adjust the methods of this paper to obtain the topological version of our main result.