MATH 74 HOMEWORK 10 Due 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 classes of loops [ γ ] and [ η ] we choose representatives γ and η so that [ γ ] * [ η ] = [ γ * η ]. If we choose diﬀerent representatives, ˜ γ and ˜ η , show that [ γ ] * [ η ] = [˜ γ * ˜ η ]. HINT: There are homotopies H 1 and H 2 between γ and ˜ γ and η and ˜ η respectively. Use these to build a homotopy between γ * η and ˜ γ * ˜ η . (3) Show that following are equivalent: (a) Every map S 1 → X is homotopic to the constant map.
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This note was uploaded on 05/04/2011 for the course MATH 73 taught by Professor Danberwick during the Spring '09 term at University of California, Berkeley.