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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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You should work it out by yourself first, but there's a nice visualization in Francis' Topological Picturebook (books.google.com/…). –  Richard Kent Jun 14 '11 at 14:16
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In fact, the isomorphism is quite far from "natural": there are choices involved, and those choices are closely linked to the braid group itself. –  Dave Futer Jun 14 '11 at 15:06
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up vote 2 down vote accepted

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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Thank you so much for the answer! Then by the same argument, if we replace the disk $D$ by some other surface (for example, a punctured torus), the conclusion is still true, right? –  Zuriel Jun 15 '11 at 10:52
    
@Zuriel: That's true. –  Jim Conant Jun 15 '11 at 13:29
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