By Monday, 26 January, please read all of Section 1.1 in Hatcher, especially the section on Induced Homomorphisms p. 34-37. (I did not finish my lecture on the 21st. What I didn’t cover was in Induced Homomorphisms).
Due Wednesday, 28 January, in class.
Select and turn in three problems from Section 1.1, p. 38-40 and from the supplementary problems A, B, and C, below.
Suggested problems: #1, 5, 8, 11, 14, 16, 17, A, B, C
Disclaimer: You should do all of the suggested problems on your own, and I will assume that you do. I would recommend (for the midterm, final, and qualifying exams) that you do many other problems. I am available for help.
A. Let \( f: (X,x) \to (Y,y) \) be a continuous map. Show that the induced map \( f_\ast: \pi_1(X, x) \to \pi_1(Y,y) \) is a group homomorphism. (I showed you the steps in class, please provide justification).
B. Let \( (X,x) \overset{f}{\to} (Y,y) \overset{g}{\to} (Z,z) \) be maps with \(y=f(x)\) and \(z=g(y)\). Show that the induced maps \( \pi_1(X,x) \overset{f_\ast}{\to} \pi_1(Y,y) \overset{g_\ast}{\to} \pi_1(Z,z) \) satisfy \(g_\ast \circ f_\ast = (g\circ f)_\ast \)
C. For any pointed space \((X,x)\), show that \((\mathrm{id}_X)_\ast = \mathrm{id}_{\pi_1(X,x)} \)
(From B and C you can deduce that a homeomorphism induces an isomorphism on the fundamental group)