On the nonrealizability of braid groups by homeomorphisms

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 $g\ge 2$.