Let
$\operatorname{Homeo}_{+}(D_{n}^{2})$
be the group of orientation-preserving homeomorphisms of
$D^{2}$
fixing the boundary pointwise and
$n$
marked points as a set. The Nielsen realization problem for the braid group asks whether the natural projection
$p_{n}:\operatorname{Homeo}_{+}(D_{n}^{2})\rightarrow B_{n}:=\unicode[STIX]{x1D70B}_{0}(\operatorname{Homeo}_{+}(D_{n}^{2}))$
has a section over subgroups of
$B_{n}$
. All of the previous methods use either torsion or Thurston stability, which do not apply to the pure braid group
$PB_{n}$
, the subgroup of
$B_{n}$
that fixes
$n$
marked points pointwise. In this paper, we show that the pure braid group has no realization inside the area-preserving homeomorphisms using rotation numbers.
Non-realizability of the pure braid group as area-preserving homeomorphisms
