On the nonrealizability of braid groups by homeomorphisms

Chen, Lei

doi:10.2140/gt.2019.23.3735

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On the nonrealizability of braid groups by homeomorphisms

Lei Chen

Geometry & Topology 23 (2019) 3735–3749

DOI: 10.2140/gt.2019.23.3735

Abstract

We show that the projection ${Homeo}_{+} (D_{n}^{2}) \to B_{n}$ does not have a section for $n \geq 6$ ; ie the braid group $B_{n}$ cannot be geometrically realized as a group of homeomorphisms of a disk fixing the boundary pointwise and $n$ marked points in the interior as a set. We also give a new proof of a result of Markovic (2007) that the mapping class group of a surface of genus $g$ cannot be geometrically realized as a group of homeomorphisms when $g \geq 2$ .

Keywords

dynamics of surfaces, braid groups, Nielsen realization

Mathematical Subject Classification 2010

Primary: 37E30, 57M60

References

Publication

Received: 26 August 2018

Revised: 2 April 2019

Accepted: 20 May 2019

Published: 30 December 2019

Proposed: Ian Agol

Seconded: Anna Wienhard, John Lott

Authors

Lei Chen
Department of Mathematics California Institute of Technology Pasadena, CA United States

Volume 23, issue 7 (2019)