Mod p p homology of unordered configuration spaces of p p points in parallelizable surfaces

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