hw11_132sp17.pdf - Math 132 Topology II Smooth Manifolds...

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Math 132 - Topology II: Smooth Manifolds. Spring 2017. Homework 11 Due in class by 1.15pm, Wednesday April 19. Late homework will not be accepted. Note: § A.x.y refers to Problem y in Section A.x in Differential Topology by Guillemin & Pollack, AMS Chelsea Pub. (2010 Edition). Reading § 3.3, 3.4 Submit solutions to the following problems: § 3.3.2 In this problem you will prove the Hairy Ball Theorem . (a) Compute the degree of the antipodal map S k S k ,x - x . (b) Prove that the antipodal map is homotopic to the identity if and only if k is odd. ( Hint: recall Problem 1.6.7 from HW4 ) (c) A point x S k is a zero of a vector field v S k TS k if v ( x ) = 0 T x S k . Show that if k is odd there exists a vector field on S k having no zeroes. A vector field admitting no zeroes is called nonvanishing . (d) Prove that if S k has a nonvanishing vector field then its antipodal map is ho- motopic to the identity id S k . (Hint: show that you can assume that v ( x ) = 1.
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