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 $\mathbb{Z} \oplus \Omega^{SO}_{2} (BG)$, where $\Omega^{SO}_{2} (BG)$ 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 $G/[G,G] \times S^1 \times X^G$, where $\pi_1 (X^G) = \Omega^{SO}_{2} (BG)$. Finally, we present some results about the classification of $G$-topological quantum field theories in dimension two.

Keywords

cobordism category, classifying space, $G$-cobordism

2010 Mathematics Subject Classification

55P91, 57R85

Full Text (PDF format)

Received 7 June 2021

Received revised 9 July 2022

Accepted 9 July 2022

Published 27 September 2023