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Unformatted text preview: . Example. In the case of onedimensional manifolds the degree of the map : M S 1 is called the winding number . Picture. Theorem 5.4.2 Let V and M be ndimensional compact oriented manifolds without boundary. Let : M V be a smooth map. Let y V be a regular value of so that the di erential * : T x M T y V at any point x 1 ( y ) is bijective (isomorphism). Then deg( ) = X x 1 ( y ) sign ( ( x )) , where sign ( ( x )) = sign (det( * )) . Proof : 1. (I). Claim: the preimage 1 ( y ) of a regular value is a nite set, that is, 1 ( y ) = { x i M  ( x i ) = y , i = 1 , 2 , . . . , N } . di geom.tex; April 12, 2006; 17:59; p. 134...
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 Spring '10
 Wong
 Geometry

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