Mod 2 cohomology of 2-configuration space of a closed surface and Stiefel--Whitney class

In this paper, we compute the singular cohomology groups $H^{\ast }(C_2(M);{\mathbb{F}}_2)$ of the ordered 2-configuration space $C_2(M)$ as ${\mathrm{\Sigma }}_2$-representations. Using the result, we determine the mod 2 cohomology of the unordered 2-configuration space $B_2(M)$ as a $H^{\ast }(\mathbb{R}{P}^{\mathrm{\infty }};{\mathbb{F}}_2)$-module. As a corollary of our computation, we see that the Stiefel–Whitney height of $M$ is $2$ or $3$ when $M$ is orientable or not, respectively.