Let $p: \mathbb{R}^{2} \rightarrow S^{1} \times S^{1}=: X$ be the map given by

$$p\left(r_{1}, r_{2}\right)=\left(e^{2 \pi i r_{1}}, e^{2 \pi i r_{2}}\right)$$

where $S^{1}$ is identified with the unit circle in $\mathbb{C}$. [You may take as given that $p$ is a covering map.]

(a) Using the covering map $p$, show that $\pi_{1}\left(X, x_{0}\right)$ is isomorphic to $\mathbb{Z}^{2}$ as a group, where $x_{0}=(1,1) \in X$.

(b) Let $\mathrm{GL}_{2}(\mathbb{Z})$ denote the group of $2 \times 2$ matrices $A$ with integer entries such that $\operatorname{det} A=\pm 1$. If $A \in \mathrm{GL}_{2}(\mathbb{Z})$, we obtain a linear transformation $A: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$. Show that this linear transformation induces a homeomorphism $f_{A}: X \rightarrow X$ with $f_{A}\left(x_{0}\right)=x_{0}$ and such that $f_{A *}: \pi_{1}\left(X, x_{0}\right) \rightarrow \pi_{1}\left(X, x_{0}\right)$ agrees with $A$ as a map $\mathbb{Z}^{2} \rightarrow \mathbb{Z}^{2}$.

(c) Let $p_{i}: \widehat{X}_{i} \rightarrow X$ for $i=1,2$ be connected covering maps of degree 2 . Show that there exist homeomorphisms $\phi: \widehat{X}_{1} \rightarrow \widehat{X}_{2}$ and $\psi: X \rightarrow X$ so that the diagram

is commutative.