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On the nonrealizability of braid groups by homeomorphisms

Lei Chen

Geometry & Topology 23 (2019) 3735–3749

We show that the projection Homeo+(Dn2) Bn does not have a section for n 6; ie the braid group Bn 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 2.

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dynamics of surfaces, braid groups, Nielsen realization
Mathematical Subject Classification 2010
Primary: 37E30, 57M60
Received: 26 August 2018
Revised: 2 April 2019
Accepted: 20 May 2019
Published: 30 December 2019
Proposed: Ian Agol
Seconded: Anna Wienhard, John Lott
Lei Chen
Department of Mathematics
California Institute of Technology
Pasadena, CA
United States