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 $H_{\ast }(B_k({\Sigma }_1);\mathbb{Z})$ has no $p$-torsion for $k\le 2p-1$. 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 $p\ge 5$. For $p\ge 5$, 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 $p$.

Comments welcome. v2: minor exposition fixes