On the non-realizability of braid groups by homeomorphisms

In this paper, we will show that the projection $$\text{Homeo}$^+(D^2_n)\to B_n$doesnothaveasection;i\mathrm{.}e\mathrm{.}thebraidgroup$B_n$ 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.