Let f (S.ro) (S,) be a continuous map. Then the induced homomorphism f. 1(S, xo) (...

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Let f (S.ro) (S,) be a continuous map. Then the induced homomorphism f. 1(S, xo) ( Z) (S,2) ( Z) is completely determined by the integer d given by f.([ao]) = d[a], where [a] is a generator (S, ) (for i = 0, 1) that represents 1 Z. This integer d is called the degree of f (denoted by deg(f)). (a) Show that if f~g, then deg(f) = deg(g). (b) Show that if f is a homeomorphism, then deg (f) = 1. In par- ticular, show that the deg(a) = -1, when a is the antipodal map. Let f (S.ro) (S,) be a continuous map. Then the induced homomorphism f. 1(S, xo) ( Z) (S,2) ( Z) is completely determined by the integer d given by f.([ao]) = d[a], where [a] is a generator (S, ) (for i = 0, 1) that represents 1 Z. This integer d is called the degree of f (denoted by deg(f)). (a) Show that if f~g, then deg(f) = deg(g). (b) Show that if f is a homeomorphism, then deg (f) = 1. In par- ticular, show that the deg(a) = -1, when a is the antipodal map.

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