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

2020/02/23 by Lei Chen, Chen, Lei
Mathematics · #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2002.10918

openalex publication_date 2020/02/23 · openalex created_date 2022/10/30 · openalex updated_date 2026/07/28

Abstract

Let Homeo₊(D²_n) be the group of orientation-preserving homeomorphisms of D² 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:Homeo₊(D²_n)→ B_n:=π₀(Homeo₊(D²_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.

Related

• Non-realizability of the Torelli group as area-preserving homeomorphisms
• Rigidity of braid group actions on ℝ and of low-genus mapping class group actions on S¹
• Geometric representations of the braid groups
• Braid groups are linear
• Braid groups are not profinitely rigid