Turn in 3 problems from Section 1.2 and the extra problems (A-D) below, and read Section 1.2 (it contains some great examples). Think about all the ways you know how to calculate \(\pi_1\) of a space.

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.

Due Wednesday, February 4.

Suggested problems:

Section 1.2 # 3, 4, 8, 10, 19

A. The (initial) maps \(\varphi_\alpha: G_\alpha \to G\) when defining the free product \(G\) of the groups \(\{G_\alpha\}_{\alpha\in \Lambda}\) are injective.

B. If the free product of groups exists, then it is unique up to isomorphism. (Use the universal property)

C. \(F(\mathcal{A}) \cong F(\mathcal{B})\) if and only if \(\#\mathcal{A} = \#\mathcal{B}\)

D. The Klein bottle \(K\) can be given a CW structure consisting of one 0-cell, two 1-cells (labelled \(a\) and \(b\)), and one 2-cell attached via the loop \(aba^{-1}b\). I claimed in class that \(K\) is homeomorphic to some orientable or nonorientable surface in the list \(\ldots N_2, N_1, M_0, M_1, M_2 \ldots\) Which one is it? Prove it is homeomorphic to \(K\). Note: this will take some creativity, and I’m happy to give a hint.

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)

Due Wednesday, 21 January, in class.

Select and turn in three problems from Chapter 0, p. 18-20.

Suggested problems: #1, 2, 3, 9, 10, 14

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.

I’m wondering how \(\LaTeX\) is working. \(\int e^x\,dx\) or\(H_1(S^1;\mathbb{Q})\)