Where can one find reference proving that Braid group induces isomorphism between punctured disk and the complement of the braid?

2011-06-14T13:04:23Z

It is a known result that if $B$ is an $n$ braid over a disk, then $B$ naturally induces an isomorphism between the fundamental group of a disk with n points removed and the fundamental group of the space $D\times [0,1]-B$, where $D$ is a disk. My question is, in which book/paper can I find a proof of this result?

Answer by Jim Conant for Where can one find reference proving that Braid group induces isomorphism between punctured disk and the complement of the braid?

2011-06-15T05:23:56Z

The isomorphism of fundamental groups comes from a diffeomorphism of spaces: $D\times[0,1]\setminus B$ is diffeomorphic to the product of an n-punctured disk with $[0,1]$. To see this, note that you can untangle the braid by sliding the ends of the braid along the surface of $D\times[0,1]$. This sliding is not a braid isotopy but it is a diffeomorphism of the complement. Indeed, you can think of a braid as an isotopy of n points in $D$, and to untangle it, you just reverse the isotopy.

Where can one find reference proving that Braid group induces isomorphism between punctured disk and the complement of the braid? - MathOverflow

Where can one find reference proving that Braid group induces isomorphism between punctured disk and the complement of the braid?

Answer by Jim Conant for Where can one find reference proving that Braid group induces isomorphism between punctured disk and the complement of the braid?