On Isotopies of Homeomorphisms of Riemann Surfaces

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'el the condition p(x) = p(x') implies pg(x) = pg(x'). If g is fiber-preserving and isotopic to the identity map via an isotopy gs, then g is said to be "fiber-isotopic to 1 " if, for every s e [0,1], the homeomorphism gs is fiber-preserving. The condition that an isotopy be a fiber-isotopy imposes a symmetry which one feels, intuitively, is very restrictive. However, we find THEOREM 1. Let g:X-> X be a fiber-preserving homeomorphism which is isotopic to the identity map. If the covering is branched, assume X is not the closed sphere or torus. Then g is fiber-isotopic to the identity. Theorem 2 expresses a weaker result, which is true without exception.