JournalsrlmVol. 18 , No. 2DOI 10.4171/rlm/486

Configuration spaces of tori

  • Yoel Feler

    Weizmann Institute of Science, Rehovot, Israel
Configuration spaces of tori cover

Abstract

The nn-point configuration spaces En(T2)={(q1,...,qn)(T2)n qiqj ij}{\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 Cn(T2)={QT2 #Q=n}{\mathcal C}^n({\mathbb T}^2) =\{Q\subset {\mathbb T}^2\,|\ \#Q=n\} of a complex torus T2{\mathbb T}^2 are complex manifolds. We prove that for n>4n>4 any holomorphic self-map FF of Cn(T2){\mathcal C}^n({\mathbb T}^2) either carries the whole of Cn(T2){\mathcal C}^n({\mathbb T}^2) into an orbit of the diagonal \Aut(T2)\Aut({\mathbb T}^2) action in Cn(T2){\mathcal C}^n({\mathbb T}^2) or is of the form F(Q)=T(Q)QF(Q)=T(Q)Q, where T ⁣:Cn(T2)\Aut(T2)T\colon{\mathcal C}^n({\mathbb T}^2)\to\Aut({\mathbb T}^2) is a holomorphic map. We also prove that for n>4n>4 any endomorphism of the torus braid group Bn(T2)=π1(Cn(T2))B_n({\mathbb T}^2)=\pi_1({\mathcal C}^n({\mathbb T}^2)) with a non-abelian image preserves the pure torus braid group Pn(T2)=π1(En(T2))P_n({\mathbb T}^2) =\pi_1({\mathcal E}^n({\mathbb T}^2)).