MATH527_HW7 - Serge Ballif MATH 527 Homework 7 October 26,...

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Unformatted text preview: Serge Ballif MATH 527 Homework 7 October 26, 2007 Problem 1. Let K = { ( x,y,z ) R 3 | x 2 + y 2 = 1 ,z = 0 } , the standard circle in R 3 . Compute the fundamental group of X = R 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 { } { } [- 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 1 ( X 2 ) = Z . Since X X 2 we know that 1 ( X ) = Z . 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 be a (non self-intersecting) path on S 2 from p to q and likewise let be a path from q to r . In X , and are loops with common point p = q = r . Let A be the subspace of X that is the union of the paths and . We adjoin a 2-cell to A to get X by define a map h : B 2 X that maps the boundary of...
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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_HW7 - Serge Ballif MATH 527 Homework 7 October 26,...

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