  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 torus are the
same as its Betti numbers for $p>2$ and $k\leq p$. Hence the integral homology
has 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 via
Lubin-Tate theory.
