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Unformatted text preview: Serge Ballif MATH 27 Homework 9 November 9, 2007 Problem 1. Let p : E B be a covering map. Assume E is path connected and B is simply connected. Show that p is a homeomorphism. Since E is path connected, the lifting correspondence : 1 ( B,b ) p- 1 ( b ) is surjective. This is true for each b B . Since 1 ( B,b ) is trivial, we know that p- 1 ( b ) consists of just a single element. Thus, p is a bijection. Also, p is continuous as a covering map and p- 1 is continuous, because p is an open map. Thus, p is a homeomorphism. Problem 2. Let G be a group of homeomorphisms of X . The action of G on X is said to be fixed-point free if no element of G other than the identity e has a fixed point. Show that if X is Hausdorff, and if G is a finite group of homeomorphisms of X whose action is fixed-point free then the action of G is properly discontinuous. Recall that the action of G is properly discontinuous if for every x X , there exists a neighborhood U of x such that g (...
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