Non-realizability of the pure braid group as area-preserving homeomorphisms

Abstract

Let Homeo+(D 2 n ) be the group of orientation-preserving homeomorphisms of D 2 fixing the boundary pointwise and n marked points as a set. Nielsen realization problem for the braid group asks whether the natural projection pn : Homeo+(D 2 n ) → Bn := π0(Homeo+(D 2 n )) has a section over subgroups of Bn. All of the previous methods either use torsions or Thurston stability, which do not apply to the pure braid group P Bn, the subgroup of Bn that fixes n marked points pointwise. In this paper, we show that the pure braid group has no realization inside the areapreserving homeomorphisms using rotation numbers.