Isotopies of homeomorphisms of Riemann surfaces and a theorem about Artin's braid group

  title={Isotopies of homeomorphisms of Riemann surfaces and a theorem about Artin's braid group},
  author={Joan S. Birman and Hugh M. Hilden},
  journal={Annals of Mathematics},
Let X, X be orientable surfaces. Let (p, X, X) be a regular covering space, possibly branched, with finitely many branch points and a finite group of covering transformations. We require also that every covering transformation leave the branch points fixed. A homeomorphism g : X -> X is said to be "fiber-preserving" with respect to the triplet (p,X,X) if for every pair of points x , x ' e l the condition p(x) = p(x') implies pg(x) = pg(x'). If g is fiber-preserving and isotopic to the identity… 
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