On the non-realizability of braid groups by homeomorphisms

TL;DR

This paper proves that braid groups cannot be realized as groups of homeomorphisms of a disk fixing the boundary and points, and provides a new proof that certain surface groups cannot be geometrically realized.

Contribution

It establishes the non-realizability of braid groups by homeomorphisms and offers a new proof for the non-realizability of surface mapping class groups.

Findings

Braid groups cannot be realized as homeomorphism groups of a disk.

A new proof is provided for the non-realizability of surface mapping class groups.

The projection from homeomorphisms to braid groups has no section.

Abstract

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.