On the non-realizability of braid groups by homeomorphisms

Abstract

In this paper, we will show that the projection ${\text{Homeo}}^{+}(D_n^2)\to B_n$ does not have a section; i.e. the braid group $B_n$ cannot be geometrically realized as a group of homeomorphisms of a disk fixing the boundary point-wise and $n$ marked points in the interior as a set. We also give a new proof of a result of Markovic that the mapping class group of a closed surface cannot be geometrically realized as a group of homeomorphisms.

13 pages, 5 figures; To appear in Geometry and Topology