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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online