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lecture36 - Lecture 36 The rst problem on todays homework...

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Lecture 36 The first problem on today’s homework will be to prove the inverse function theorem for manifolds. Here we state the theorem and provide a sketch of the proof. Let X, Y be n -dimensional manifolds, and let f : X Y be a C map with f ( p ) = p 1 . Theorem 6.39. If df p : T p X T p 1 Y is bijective, then f maps a neighborhood V of p diffeomorphically onto a neighborhood V 1 of p 1 . Sketch of proof: Let φ : U V be a parameterization of X at p , with φ ( q ) = p . Similarly, let φ 1 : U 1 V 1 be a parameterization of Y at p 1 , with φ 1 ( q 1 ) = p 1 . Show that we can assume that f : V V 1 (Hint: if not, replace V by V f 1 ( V 1 )). Show that we have a diagram f V V 1 −−−→ (6.114) φ φ 1 g −−−→ U U 1 , which defines g , g = φ 1 1 f φ, (6.115) g ( q ) = q 1 . (6.116) So, ( dg ) q = ( 1 ) 1 q 1 df p ( ) q . (6.117) Note that all three of the linear maps on the r.h.s. are bijective, so ( dg ) q is a bijection. Use the Inverse Function Theorem for open sets in R n . This ends our explanation of the first homework problem. Last time we showed the following. Let X, Y be n -dimensional manifolds, and let f : X Y be a proper C map. We can define a topological invariant deg( f ) such that for every ω Ω n c ( Y ), f
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