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?
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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. |
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