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

TL;DR

This paper proves that the pure braid group cannot be realized as a subgroup of area-preserving homeomorphisms of a disk fixing points, using rotation numbers to establish this non-realizability.

Contribution

It demonstrates the non-realizability of the pure braid group within area-preserving homeomorphisms, filling a gap left by previous methods that do not apply to pure braids.

Findings

Pure braid group cannot be embedded into area-preserving homeomorphisms.

Rotation numbers are used to establish non-realizability.

Extends understanding of Nielsen realization problem for pure braids.

Abstract

Let $\text{Homeo}_+(D^2_n)$ 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 $p_n:\text{Homeo}_+(D^2_n)\to B_n:=\pi_0(\text{Homeo}_+(D^2_n))$ has a section over subgroups of $B_n$. All of the previous methods either use torsions 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.