prac2_132sp17.pdf - Math 132 Topology II Smooth Manifolds...

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Math 132 - Topology II: Smooth Manifolds. Spring 2017. Week 2 Practice Problems/Exercises These problems/exercises are not to be submitted and are supposed to get you think- ing about the material. Some are very straightforward and some may require some more thought. They should be considered supplementary to lecture. Lecture 4 1. Show that T ( R k ) = R k × R k . 2. Let f X Y be a smooth map. Show that df TX TY is smooth. 3. Let f X Y , g Y Z be smooth. Show that d ( g f ) = dg df . 4. Show that TS n R n + 1 × R n + 1 is equal to {( x,y ) R n + 1 × R n + 1 x y = 0 } where x y is the usual dot product. 5. Give an example of a vector field v S 1 TS 1 such that v ( p ) 0, for all p S 1 (we call such a vector field nonvanishing ). Can you generalise your example to construct a nonvanishing vector field on S n , where n is odd? (We will see later in this course that it is impossible to find nonvanishing vector fields on spheres S n with n even) 6. Prove that a closed ball B a ( x ) = { y R k
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