Mod 2 representation of the symmetric group of order 2 over cohomology groups of 2-configuration space of torus

In this article, we compute the mod 2 representarion of the symmetric group of order 2 over the singular cohomology groups of orderd 2-configuration space $C_2(T^d)$ of the $d$-torus $T^d$ for $d\ge 1$. As applications of the computation, we determine the Stiefel-Whitney height of $C_2(T^d)$ for any $d$, and determine ${\mathbb{F}}_2[{\mathrm{\Sigma }}_2]$-module structure of the cohomology groups of the unordered 2-configuration space of the $d$-torus for $d=2,3$ using the Serre spectral sequence.