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Unformatted text preview: ϕ . • Example. • In the case of onedimensional manifolds the degree of the map ϕ : 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 ϕ : M → V be a smooth map. Let y ∈ V be a regular value of ϕ so that the di ﬀ erential ϕ * : T x M → T y V at any point x ∈ ϕ1 ( y ) is bijective (isomorphism). Then deg( ϕ ) = X x ∈ ϕ1 ( y ) sign ( ϕ ( x )) , where sign ( ϕ ( x )) = sign (det( ϕ * )) . Proof : 1. (I). Claim: the preimage ϕ1 ( y ) of a regular value is a ﬁnite set, that is, ϕ1 ( y ) = { x i ∈ M  ϕ ( x i ) = y , i = 1 , 2 , . . . , N } . di ﬀ geom.tex; April 12, 2006; 17:59; p. 134...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry

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