hw5-difftop

# hw5-difftop - Math 343 Homework 5 Due 4pm Friday I will be...

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Math 343 Homework 5 Due 4pm Friday March 12, 2010. I will be away at MSRI in Berkeley CA from March 7–19. We can set up times to discuss this homework via Skype while I am away. The homework will be collected for me at 4pm on the day it is due. 1 To be handed in for grading 1. Which of the following linear spaces intersect transversally? Explain your answers. (a) xy -plane and the z -axis in R 3 (b) R k × { 0 } and { 0 } × R l in R n . (Depends on k,l,n .) (c) V × { 0 } and the diagonal in V × V 2. Transversality and vector spaces. (a) Suppose V and W are vector subspaces of R n . The space V + W is deﬁned as follows V + W = { ~v + ~w | ~v V, ~w W } . Check that V + W is a vector subspace of R n . (b) Check that V t W means just V + W = R n . 3. If U R k and V H k are neighborhoods of 0, prove that there exists no diﬀeomor- phism of V with U . 4. Prove that if f : X Y is a diﬀeomorphism of manifolds with boundary, then ∂f maps ∂X diﬀeomorphically onto ∂Y . (Recall

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hw5-difftop - Math 343 Homework 5 Due 4pm Friday I will be...

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