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

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Unformatted text preview: Serge Ballif MATH 527 Homework 8 November 2, 2007 Problem 1. (a) Show that every continuous map f : R 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 R P 2 is a quotient space of S 2 via a quotient map q . Moreover, the map q 1 I : S 2 I R P 2 I is a quotient map, because q and 1 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 1 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 } is a continuous map. Moreover, p is not nulhomotopic because, p * maps 1 ( S 1 S 1 ) onto 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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This note was uploaded on 04/01/2008 for the course MATH 527 taught by Professor Rotman,regina during the Fall '07 term at Pennsylvania State University, University Park.

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

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