The classifying space of the 1+1 dimensional $G$-cobordism category

The 1+1 G-cobordism category, with G a finite group, is important in the construction of G-topological field theories which are completely determined by a G-Frobenius algebra. We give a description of the classifying space of this category generalizing the work of Ulrike Tillmann. Moreover, we compute the connected components and the fundamental group of this classifying space and we give a complete description of the classifying spaces of some important subcategories. Finally, we present some relations between the rank of the fundamental group of the G-cobordism category and the number of subgroups of the group G.