On the non-realizability of braid groups by homeomorphisms

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