On the Classifying Space of Some Cobordism Categories

Abstract

In this dissertation, the classifying spaces of some cobordism categories are investigated. Specifically, it is proved that the classifying space of the category of open strings is homotopy equivalent with the classifying space of the category of open and closed strings as studied by Baas, Cohen and Ramírez. The proof is similar to the original one. One point of note is that the category considered here is based on the atomic surface model as first introduced by Tillmann. This is a 2-category that is combinatorial in nature and somewhat easier to work with than the category utilised by Baas, Cohen and Ramírez.

Using these atomic surfaces, a comparison of open and closed strings using various models for the category of open strings and the category of closed strings is then presented. The homotopy type of the categories presented here is dependent on whether \windows" are allowed or not, and not whether the strings are closed or open or both.

In a joint work with Elizabeth Hanbury, a conjecture due to Baas about classifying spaces of n-dimensional cobordism categories with marked n-discs is investigated. This conjecture relates to a toy example of cobordism categories with singularities. It is a higher-dimensional analogue of the theorem due to Baas, Cohen and Ramírez determining the classifying space of the category of open and closed strings. Evidence in favour of the conjecture is presented using a direct approach of computing the classifying space of the category.