lecture29 - Lecture 29 We have been studying the important...

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Lecture 29 We have been studying the important invariant called the degree of f . Today we show that the degree is a “topological invariant.” 5.3 Topological Invariance of Degree Recall that given a subset A of R m and a function F : A R , we say that F is C if it extends to a C map on a neighborhood of A . Let U be open in R n , let V be open in R k , and let A = U × [0 , 1]. Definition 5.22. Let f 0 , f 1 : U V be C maps. The maps f 0 and f 1 are homotopic if there is a C map F : U × [0 , 1] V such that F ( p, 0) = f 0 ( p ) and F ( p, 1) = f 1 ( p ) for all p U . Let f t : U V be the map defined by f t ( p ) = F ( p, t ) . (5.144) Note that F C = So, f t : U V , where 0 t 1, gives a family f t C . of maps parameterized by t . The family of maps f t is called a C deformation of f 0 into f 1 . Definition 5.23. The map F is a proper homotopy if for all compact sets A V , the pre-image F 1 ( A ) is compact. Denote by π the map π : U × [0 , 1] U that sends ( p, t ) t . Let A V be compact. Then B = π ( F 1 ( A )) is compact, and for all t , f 1 ( A ) B . As a t consequence, each f t is proper.
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