Mod $p$ homology of unordered configuration spaces of surfaces

TL;DR

This paper proves that for certain surfaces, the mod p homology of unordered configuration spaces matches Betti numbers, indicating no p-torsion in integral homology, using a simplified proof applicable to various surfaces.

Contribution

It provides a concise proof that the mod p homology of configuration spaces on surfaces aligns with Betti numbers for specific parameters, extending previous results.

Findings

Homology groups match Betti numbers for p>2 and k≤p.

Integral homology has no p-power torsion.

Method applies to punctured genus g surfaces, recovering known results.

Abstract

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.