The classifying space of the $G$-cobordism category in dimension two

Carlos Segovia

Published 2012-11-09, updated 2019-03-04Version 3

For $G$ a finite group, we define the $G$-cobordism category in dimension two. We prove this category has connected components in bijection with the abelianization of $G$ and with fundamental group isomorphic to a lattice. The main result consists of an splitting of the classifying space of the $G$-cobordism category as the product of the abelianization of $G$, an infinite loop space and a finite product of circles. In addition, we study the classifying space of some important subcategories.