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 C₂(Tᵈ) of the d-torus Tᵈ for d≥ 1. As applications of the computation, we determine the Stiefel-Whitney height of C₂(Tᵈ) for any d, and determine F₂[Σ₂]-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.