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Session S27 - Categories and Topology

Wednesday, July 21, 18:00 ~ 18:30 UTC-3

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

Carlos Segovia González

Instituto de Matemáticas UNAM-Oaxaca, México   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

For $G$ a finite group, we define the free $G$-cobordism category in dimension two. We show the classifying space of this category has connected components in bijection with the abelianization of $G$ and with fundamental group isomorphic to the direct sum $\mathbb{Z}+H_2(G)$, where $H_2(G)$ is the integral 2-homology group. For $G$ a finite abelian group, we study its classifying space showing an splitting as $G\times X^G\times T^{r(G)}$, where $X^G$ is a simply connected infinite loop space and $T^{r(G)}$ is the product of $r(G)$ circles. An explicit expression for the number $r(G)$ is presented. Also, a description of the classifying space of some important subcategories is provided. Finally, we present some results relative to the classification of $G$-topological quantum field theories in dimension two.

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