## View abstract

### 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. document.getElementById('cloak027e5ef6ffdc906e13bda11e844ab27e').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy027e5ef6ffdc906e13bda11e844ab27e = 'cs&#101;g&#111;v&#105;&#97;' + '&#64;'; addy027e5ef6ffdc906e13bda11e844ab27e = addy027e5ef6ffdc906e13bda11e844ab27e + 'm&#97;t&#101;m' + '&#46;' + '&#117;n&#97;m' + '&#46;' + 'mx'; var addy_text027e5ef6ffdc906e13bda11e844ab27e = 'cs&#101;g&#111;v&#105;&#97;' + '&#64;' + 'm&#97;t&#101;m' + '&#46;' + '&#117;n&#97;m' + '&#46;' + 'mx';document.getElementById('cloak027e5ef6ffdc906e13bda11e844ab27e').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy027e5ef6ffdc906e13bda11e844ab27e + '\'>'+addy_text027e5ef6ffdc906e13bda11e844ab27e+'<\/a>';

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.