Configuration spaces of tori

Abstract

The $n$-point configuration spaces ${\mathcal{E}}^n({\mathbb{T}}^2)=\{(q_1,...,q_n)\in ({\mathbb{T}}^2{)}^n|q_i\ne q_j\forall i\ne j\}$ and ${\mathcal{C}}^n({\mathbb{T}}^2)=\{Q\subset {\mathbb{T}}^2|\#Q=n\}$ of a complex torus ${\mathbb{T}}^2$ are complex manifolds. We prove that for $n>4$ any holomorphic self-map $F$ of ${\mathcal{C}}^n({\mathbb{T}}^2)$ either carries the whole of ${\mathcal{C}}^n({\mathbb{T}}^2)$ into an orbit of the diagonal $\mathrm{Aut}({\mathbb{T}}^2)$ action in ${\mathcal{C}}^n({\mathbb{T}}^2)$ or is of the form $F(Q)=T(Q)Q$, where $T:{\mathcal{C}}^n({\mathbb{T}}^2)\to \mathrm{Aut}({\mathbb{T}}^2)$ is a holomorphic map. We also prove that for $n>4$ any endomorphism of the torus braid group $B_n({\mathbb{T}}^2)={\pi }_1({\mathcal{C}}^n({\mathbb{T}}^2))$ with a non-abelian image preserves the pure torus braid group $P_n({\mathbb{T}}^2)={\pi }_1({\mathcal{E}}^n({\mathbb{T}}^2))$.