Cobordism categories of manifolds with corners

TL;DR

This paper determines the homotopy types of cobordism categories of manifolds with corners, extending previous results and applying to categories with additional tangential structures and string backgrounds.

Contribution

It generalizes the homotopy type identification of cobordism categories to manifolds with corners and structured tangential data, building on and extending prior foundational work.

Findings

Identifies the homotopy type of cobordism categories with corners as zero spaces of homotopy colimits of Thom spectra.

Extends results to categories with extra tangential structures.

Describes the homotopy type of string categories with background space X.

Abstract

In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if {Cob}_{d,<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 B{Cob}_{d,<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.