Cobordism categories of manifolds with corners

Abstract

In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if Cob d , ⟨ k ⟩ \text {Cob}_{d,\langle k\rangle } is the category whose objects are a fixed dimension d d , with corners of codimension ≤ k \leq \;k , then we identify the homotopy type of the classifying space B Cob d , ⟨ k ⟩ B\text {Cob}_{d,\langle k\rangle } as the zero space of a homotopy colimit of a certain diagram of the Thom spectra. We also identify the homotopy type of the corresponding cobordism category when an extra tangential structure is assumed on the manifolds. These results generalize the results of Galatius, Madsen, Tillmann and Weiss (2009), and their proofs are an adaptation of the methods of their paper. As an application we describe the homotopy type of the category of open and closed strings with a background space X X , as well as its higher dimensional analogues. This generalizes work of Baas-Cohen-Ramirez (2006) and Hanbury.