MATH 74 HOMEWORK 10Due Wednesday May 6th at 3:10pm.(1) Prove that homotopy defines an equivalence relation on the set of based loops in a topological space.(2) Prove that multiplication in the fundamental group is well-defined. That is, given equivalence classesof loops [γ] and [η] we choose representativesγandηso that [γ]*[η] = [γ*η]. If we choose differentrepresentatives, ˜γand ˜η, show that [γ]*[η] = [˜γ*˜η].HINT: There are homotopiesH1andH2betweenγand ˜γandηand ˜ηrespectively. Use these to build a homotopy betweenγ*ηand ˜γ*˜η.(3) Show that following are equivalent:(a) Every mapS1→Xis homotopic to the constant map.
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