Construct a space $X$ as follows. Let $Z_{1}, Z_{2}, Z_{3}$ each be homeomorphic to the standard 2-sphere $S^{2} \subseteq \mathbb{R}^{3}$. For each $i$, let $x_{i} \in Z_{i}$ be the North pole $(1,0,0)$ and let $y_{i} \in Z_{i}$ be the South pole $(-1,0,0)$. Then

$$X=\left(Z_{1} \sqcup Z_{2} \sqcup Z_{3}\right) / \sim$$

where $x_{i+1} \sim y_{i}$ for each $i$ (and indices are taken modulo 3 ).

(a) Describe the universal cover of $X$.

(b) Compute the fundamental group of $X$ (giving your answer as a well-known group).

(c) Show that $X$ is not homotopy equivalent to the circle $S^{1}$.