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We provide a short proof that the dimensions of the mod $p$ homology groupsof the unordered configuration space $B_k(T)$ of $k$ points in a torus are thesame as its Betti numbers for $p>2$ and $k\leq p$. Hence the integral homologyhas no $p$-power torsion. The same argument works for the punctured genus $g$surface with $g>0$, thereby recovering a result of Brantner-Hahn-Knudsen viaLubin-Tate theory.

Mod $p$ homology of unordered configuration spaces of surfaces