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# ch5 - Chapter Five Cauchys Theorem 5.1 Homotopy Suppose D...

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Chapter Five Cauchy’s Theorem 5.1. Homotopy. Suppose D is a connected subset of the plane such that every point of D is an interior point—we call such a set a region —and let C 1 and C 2 be oriented closed curves in D . We say C 1 is homotopic to C 2 in D if there is a continuous function H : S D , where S is the square S  t , s : 0 s , t 1 , such that H t ,0 describes C 1 and H t ,1 describes C 2 , and for each fixed s , the function H t , s describes a closed curve C s in D . The function H is called a homotopy between C 1 and C 2 . Note that if C 1 is homotopic to C 2 in D , then C 2 is homotopic to C 1 in D . Just observe that the function K t , s H t ,1 s is a homotopy. It is convenient to consider a point to be a closed curve. The point c is a described by a constant function t c . We thus speak of a closed curve C being homotopic to a constant—we sometimes say C is contractible to a point. Emotionally, the fact that two closed curves are homotopic in D means that one can be continuously deformed into the other in D . Example Let D be the annular region D z : 1 | z | 5 . Suppose C 1 is the circle described by 1 t 2 e i 2 t , 0 t 1; and C 2 is the circle described by 2 t 4 e i 2 t , 0 t 1. Then H t , s 2 2 s e i 2 t is a homotopy in D between C 1 and C 2 . Suppose C 3 is the same circle as C 2 but with the opposite orientation; that is, a description is given by 3 t 4 e i 2 t , 0 t 1. A homotopy between C 1 and C 3 is not too easy to construct—in fact, it is not possible! The moral: orientation counts. From now on, the term ”closed

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ch5 - Chapter Five Cauchys Theorem 5.1 Homotopy Suppose D...

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