THE CLASSIFYING SPACE OF THE 1+1 DIMENSIONAL G-COBORDISM CATEGORY

For a finite group G, we define the G-cobordism category in dimension two. We show there is a one-to-one correspondence between the connected components of its classifying space and the abelianization of G. Also, we find an isomorphism of its fundamental group onto the direct sum Z (R) ohm S2 degrees (BG), where ohm S degrees 2 (BG) is the free oriented G-bordism group in dimension two, and we study the classifying space of some important subcategories. We obtain the classifying space has the homotopy type of the product G/[G, G] x S1 x XG, where pi 1(XG) = ohm S degrees 2 (BG). Finally, we present some results about the classification of G topological quantum field theories in dimension two.