Where can one find reference proving that Braid group induces isomorphism between punctured disk and the complement of the braid? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:56:39Z http://mathoverflow.net/feeds/question/67757 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67757/where-can-one-find-reference-proving-that-braid-group-induces-isomorphism-between Where can one find reference proving that Braid group induces isomorphism between punctured disk and the complement of the braid? Zuriel 2011-06-14T13:04:23Z 2011-06-15T05:23:56Z <p>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?</p> http://mathoverflow.net/questions/67757/where-can-one-find-reference-proving-that-braid-group-induces-isomorphism-between/67835#67835 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? Jim Conant 2011-06-15T05:23:56Z 2011-06-15T05:23:56Z <p>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.</p>