MATH527_HW6 - Serge Ballif MATH 527 Homework 6 Problem 1...

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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 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 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 = D 2 D 2 is path connected. The space D 2 is contractible, so S 2 D 2 S 2 . Therefore,
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