We show that the projection does not have a section for ; ie the braid group cannot be geometrically realized as a group of homeomorphisms of a disk fixing the boundary pointwise and 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 cannot be geometrically realized as a group of homeomorphisms when .
"On the nonrealizability of braid groups by homeomorphisms." Geom. Topol. 23 (7) 3735 - 3749, 2019. https://doi.org/10.2140/gt.2019.23.3735