hw5_132sp17.pdf - Math 132 Topology II Smooth Manifolds...

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Math 132 - Topology II: Smooth Manifolds. Spring 2017. Homework 5 Due in class by 1.15pm, Wednesday March 1. 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 § 1.8, 2.2, 1.5, 2.3 Submit solutions to the following problems: § 1.8.10 Prove that every k -dimensional manifold X admits an immersion f X R 2 k § 1.8.11 Show that if X is a compact k -dimensional manifold, then there exists a map X R 2 k - 1 that is an immersion except at finitely many points of X . Do so by showing that if f X R 2 k is an immersion and a is a regular value for the map F TX R 2 k , F ( x,v ) = ( df ) x ( v ) , then F - 1 ( a ) is a finite set. Show that π f is an immersion except on f - 1 ( a ) , where π is the orthogonal projection along a . ( Hint: show that there are only finitely many preimages of a under F in the compact set {( x,v ) v 1 } TX .
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