Prove that two maps of the circle S¹ into itself are homotopic if and only if they have the same degree. This is a special case of a remarkable theorem of Hopf, which we will prove later. [HINT: If go, g₁ : R¹ R¹ both satisfy g(t + 1) = g(t) + 2лq, then so do all the maps g, = $8₁ + (1 - s)go.]
Prove that two maps of the circle S¹ into itself are homotopic if and only if they have the same degree. This is a special case of a remarkable theorem of Hopf, which we will prove later. [HINT: If go, g₁ : R¹ R¹ both satisfy g(t + 1) = g(t) + 2лq, then so do all the maps g, = $8₁ + (1 - s)go.]
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 20E: Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ]...
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Please only use definitions, propositions, theorems given in the book ''Differential Topology'' by Guillemin and Pollack !
And if you refer to a proposition, theorem or exercise in the book please refer to them by chapter and section!
Also please explain each step, even if it seems trivial to you!
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