Posted: January 8th, 2017

Let x0 = (1, 0), which is the identity element for the operation +S1 , and let f, g : S 1 → S 1 be such that f(x0) = g(x0) = x0. Show that (f +S1 g)∗ = (f∗ ? g∗) : π1(S 1 , x0) → π1(S 1 , x0).

3. Let f : S 1 → S 1 be a map such that f∗ : π1(S 1 , x0) → π1(S 1 , x0) is the zero homomorphism. Show that f is null.

4. Recall the notation +S1 for addition of points in S 1 as in problem set 8, question 4 b). Let x0 = (1, 0), which is the identity element for the operation +S1 , and let f, g : S 1 → S 1 be such that f(x0) = g(x0) = x0. Show that (f +S1 g)∗ = (f∗ ? g∗) : π1(S 1 , x0) → π1(S 1 , x0). (Here, by definition, (f +S1 g)(x) = f(x) +S1 g(x) and (f∗ ? g∗)(α) = f∗(α) ? g∗(α).)