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LectureNotes10G - Math 598 Geometry and Topology II Spring...

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Math 598 Mar 20, 2005 1 Geometry and Topology II Spring 2005, PSU Lecture Notes 10 3.4 Proof of Sard’s Theorem This section will be typeset later. In the meantime the reader is refered to Milnor’s book on Topology from Differentiable View Point . Exercise 1. Show that Sard theorem immediately implies that if f : M N is a smooth function and dim( M ) < dim( N ) then f ( M ) has measure zero. Exercise 2. Use Sard’s theorem to show that S n is simply connected for n 2. (Hint: it is enough to show that every continuos map f : S 1 S 2 is homotopic to a map f : S 1 S 2 which is not onto. You also need to use Wierstrauss’s approximation theorem.) Exercise 3. Let M n be a compact manifold smoothly embedded in
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