2008/10/03 by Josh Genauer, Genauer, Josh · 1 citation
Mathematics · #55N22 #55P47 #57R19 #57R90 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:55N22 #msc:55P47 #msc:57R19 #msc:57R90
paper · pdf · doi:10.48550/arxiv.0810.0581
43 pages, 6 figures
arxiv created 2008/10/03 · openalex publication_date 2008/10/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if Cobd,<k> is the category whose objets are a fixed dimension d, with corners of codimension less than or equal to k, then we identify the homotopy type of the classifying space BCobd,<k> as the zero space of a homotopy colimit of certain diagram of Thom spectra. We also identify the homotopy type of the corresponding cobordism category when extra tangential structure is assumed on the manifolds. These results generalize the results of Galatius, Madsen, Tillmann and Weiss, and the proofs are an adaptation of the their methods. As an application we describe the homotopy type of the category of open and closed strings with a background space X, as well as its higher dimensional analogues. This generalizes work of Baas-Cohen-Ramirez and Hanbury.