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

@article{Birman1972IsotopiesOH, 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}, year={1972}, volume={97}, pages={424-439} }

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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## References

SHOWING 1-10 OF 13 REFERENCES

Geometrical realization of isomorphisms between plane groups

- Mathematics
- 1965

az + b w = ; a, b, c, d real ad — be = — 1. cz + d We denote the upper half-plane by D, so that [G, D] is a transformation group with the topology on G and D induced by the parameters a, b, c, d, z.…

Introduction to Riemann Surfaces

- Mathematics
- 1957

Introduction: 1-1 Algebraic functions and Riemann surfaces 1-2 Plane fluid flows 1-3 Fluid flows on surfaces 1-4 Regular potentials 1-5 Meromorphic functions 1-6 Function theory on a torus General…

Combinatorial group theory. Presentations of groups in terms of generators and relations. Second rev

- Mathematics
- 1966

Die Klassen von topologischen Abbildungen einer geschlossenen Fläche auf sich

- Mathematics, Philosophy
- 1939

Die topologischen Abbildungen einer geschlossenen orientierbaren oder nichtorientierbaren Flache F auf sich zerfallen bekanntlich in Isotopieklassen. Zwei Abbildungen gehoren zu derselben…