Odd-primary torsion in the homology of unordered configurations on the torus

Abstract

Let $B_k(\Sigma_1)$ denote the unordered configuration space of $k$ points in the torus $\Sigma_1$. For every odd prime $p$, we prove that $H_*(B_k(\Sigma_1);\mathbb{Z})$ has no $p$-torsion for $k\leq 2p-1$. At the threshold $k=2p$, we prove that $H_{2p-2}(B_{2p}(\Sigma_1);\mathbb{Z})$ has $p$-torsion if and only if $p\geq 5$. For $p\geq5$, the class is the image under puncture filling of the unique Bianchi–Stavrou order-$p$ class on the once-punctured torus; for $p=3$, Napolitano’s calculation shows that this punctured class dies after filling. We also prove that $H_{2p}(B_{2p}(\Sigma_1);\mathbb{Z})$ and $H_{2p+1}(B_{2p}(\Sigma_1);\mathbb{Z})$ have no $p$-torsion for every odd $p$.