#### Volume 23, issue 7 (2019)

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On the nonrealizability of braid groups by homeomorphisms

### Lei Chen

Geometry & Topology 23 (2019) 3735–3749
##### Abstract

We show that the projection ${Homeo}_{+}\left({D}_{n}^{2}\right)\to {B}_{n}$ does not have a section for $n\ge 6$; 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 $g\ge 2$.

##### Keywords
dynamics of surfaces, braid groups, Nielsen realization
##### Mathematical Subject Classification 2010
Primary: 37E30, 57M60