MATH527_HW8 - Serge Ballif MATH 527 Homework 8 November 2...

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Serge Ballif MATH 527 Homework 8 November 2, 2007 Problem 1. (a) Show that every continuous map f : P 2 S 1 is nulhomotopic. (b) Find a continuous map of the torus into S 1 that is not nulhomotopic. (a) Suppose that f is homotopic to a nontrivial loop in S 1 via a homotopy F . We know that P 2 is a quotient space of S 2 via a quotient map q . Moreover, the map q × I : S 2 × I P 2 × I is a quotient map, because q and I are quotient and I is locally compact Hausdorff. Define a map g : S 2 S 1 by g = f q . Then g is continuous as a composition of continuous maps. Then F ( q I ) is a homotopy between g and a nontrivial element of S 1 . This contradicts the fact that every map from S n to S 1 is nulhomotopic for n > 1. (b) The projection p of the torus S 1 × S 1 onto its first coordinate S 1 × { x 0 } is a continuous map. Moreover, p is not nulhomotopic because, p * maps π 1 ( S 1 × S 1 ) onto π 1 ( S 1 ). Problem 2. Describe the universal covering space of the following spaces: (a) the Klein bottle; (b) the n -fold Dunce Cap; (c) the figure 8.
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