(a) Let $f: X \rightarrow Y$ be a map of spaces. We define the mapping cylinder $M_{f}$ of $f$ to be the space

$$(([0,1] \times X) \sqcup Y) / \sim$$

with $(0, x) \sim f(x)$. Show carefully that the canonical inclusion $Y \hookrightarrow M_{f}$ is a homotopy equivalence.

(b) Using the Seifert-van Kampen theorem, show that if $X$ is path-connected and $\alpha: S^{1} \rightarrow X$ is a map, and $x_{0}=\alpha\left(\theta_{0}\right)$ for some point $\theta_{0} \in S^{1}$, then

$$\pi_{1}\left(X \cup_{\alpha} D^{2}, x_{0}\right) \cong \pi_{1}\left(X, x_{0}\right) /\langle\langle[\alpha]\rangle\rangle$$

Use this fact to construct a connected space $X$ with

$$\pi_{1}(X) \cong\left\langle a, b \mid a^{3}=b^{7}\right\rangle$$

(c) Using a covering space of $S^{1} \vee S^{1}$, give explicit generators of a subgroup of $F_{2}$ isomorphic to $F_{3}$. Here $F_{n}$ denotes the free group on $n$ generators.