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

Abstract

The main object of study of this paper is $B_k(\Sigma_1)$, the unordered configuration space of $k$ points in the torus $\Sigma_1$. First, we produce a marked-point transfer argument which, combined with Chen–Zhang’s theorem, establishes that $H_*(B_k(\Sigma_1);\mathbb{Z})$ has no $p$-torsion for $k\leq 2p-1$. Secondly, 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 a generator of the Bianchi–Stavrou torsion group $\mathbb{Z}/p$ of the once-punctured torus; for $p=3$, Napolitano’s calculation shows that this punctured class dies after filling. Finally, 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 $p$.