MOD p HOMOLOGY OF UNORDERED CONFIGURATION SPACES OF 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(T) of k points in a closed torus are the same as its Betti numbers for p > 2 and k ≤ 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 ≥ 0, thereby recovering a result of Brantner-Hahn-Knudsen via Lubin-Tate theory.