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

Abstract

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