MATH527_HW7 - Serge Ballif Problem 1 Let K =(x y z group of...

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Serge Ballif MATH 527 Homework 7 October 26, 2007 Problem 1. Let K = { ( x, y, z ) 3 | x 2 + y 2 = 1 , z = 0 } , the standard circle in 3 . Compute the fundamental group of X = 3 - K . X deformation rectracts onto a space X 1 which is sphere centered at the origin together with a line segment joining the North and South poles. That is X X 1 = S 2 ∪ { 0 } × { 0 } × [ - 1 , 1]. X 1 is homotopic to a space X 2 which is formed by contracting the line segment of X 1 to a point. That is, X 2 is sphere with two opposite poles identified. In class we showed that π 1 ( X 2 ) = . Since X X 2 we know that π 1 ( X ) = . Problem 2. Let X be the topological space obtained from S 2 by identifying 3 distinct points X = S 2 /p, q, r . Find the fundamental group of X . Let α be a (non self-intersecting) path on S 2 from p to q and likewise let β be a path from q to r . In X , α and β are loops with common point p = q = r . Let A be the subspace of X that is the union of the paths α and β . We adjoin a 2-cell to A to get X by define a map h : B 2 X that maps the boundary of S 1 of B 2 into A via the assignment αβ β α . The map i * : π 1 ( A, p
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