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Unformatted text preview: Serge Ballif MATH 527 Homework 6 October 12, 2007 Problem 1. Does the Borsuk-Ulam theorem hold for the torus? In other words, for every map f : S 1 S 1 R 2 must there exist ( x,y ) S 1 S 1 , such that f ( x,y ) = f (- x,- y ) ? No. We can imbed the torus in R 3 symmetric about the z-axis and the xy-plane. Then the projection from torus to an annulus in the plane ( f ( x,y,z ) = ( x,y )) is a continuous map. Since the map f preserves the angles in the xy-plane the antipodal points of the torus are mapped to what we might call antipodal points of the annulus. Thus, f ( x,y ) 6 = f (- x,- y ). Problem 2. Compute the fundamental group of S 2 S 2 . Let X = S 2 S 2 with common point p . If we remove any point other than p from X we get an open subspace space homeomorphic to S 2 D 2 where D 2 is a closed two dimensional disk. We can express X as the union of two such subspaces: U 1 = S 2 D 2 and U 2 = D 2 S 2 . The intersection U 1 U 2 =...
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This note was uploaded on 04/01/2008 for the course MATH 527 taught by Professor Rotman,regina during the Spring '07 term at Pennsylvania State University, University Park.
- Spring '07