MATH527_HW9 - Serge Ballif MATH 27 Homework 9 November 9...

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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 φ : π 1 ( B, b 0 ) p - 1 ( b 0 ) is surjective. This is true for each b 0 B . Since π 1 ( B, b 0 ) is trivial, we know that p - 1 ( b 0 ) 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 ( U ) U = when g 6 = e . Suppose that X
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