Due Friday, October 30

3.2.3, 3.3.5 through 3.3.8, From Section 3.11: #11, 12, 22

A. If \(f:(X,x) \to (Y,y)\) is a continuous map of pointed topological spaces, verify that \(f_\ast: \pi_1(X,x) \to \pi_1(Y,y)\) is a group homomorphism. I showed you the steps in class. Justify them.

B. If \(f:(X,x) \to (Y,y)\) and \(g:(Y,y) \to (Z,z)\), verify that \(g_\ast \circ f_\ast = (g\circ f)_\ast\).

C. If \(f:(X,x) \to (X,x)\) is the identity map, prove \(f_\ast\) is the identity map on the fundamental group.

I will select some problems from those to grade. Please ask me about any questions you have on the problems that are not graded.