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.