sol-11 - MATH 453 SOLUTIONS TO ASSIGNMENT 11 DECEMBER 3,...

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M  453 S   A  11 D  3, 2004 Exercise 2 from Section 51, page 330 This is a special case of problem 3 below. ± Exercise 3 from Section 51, page 330 (a) The formula F ( x , t ) = (1 - t ) x is a homotopy between the identity map on either I or R to the constant map at 0. (b) Suppose X is contractible and let F : X × I X be a nullhomotopy with F ( x , 0) = x and F ( x , 1) = x 0 for all x . If x , y X , F ( x , t ) and F ( y , t ) are paths from x and y to x 0 respectively. Then F ( x , t ) * F ( y , t ) is a path from x to y . (c) Y is contractible, so i Y ' e y 0 where e y 0 is the constant map at y 0 . Thus given any map f : X Y , we have i Y f ' e y 0 f . Since i Y f = f , we see that any map is homotopic to the constant map that takes all of X to y 0 . (d) Let i X ' e x 0 and [ f ] , [ g ] [ X , Y ]. Then f = f i X ' f e x 0 = e f ( x 0 ) and similarly g ' e g ( x 0 ) . Hence we need only show that
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sol-11 - MATH 453 SOLUTIONS TO ASSIGNMENT 11 DECEMBER 3,...

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