 Solution

Homotopy equivalence means maps $f:X\to Y$, $g:Y\to X$ with $gf\simeq {\mathrm{id}}_X$ and $fg\simeq {\mathrm{id}}_Y$. Radial retraction of ${\mathbb{C}}^{\ast }$ and contraction of $\mathbb{C}$ give ${\mathbb{C}}^{\ast }\times \mathbb{C}\simeq S^1$. Coordinates $(a,b)\mapsto (a-b,b)$ identify $X\simeq {\mathbb{C}}^{\ast }\times \mathbb{C}$, so ${\pi }_1(X)=\mathbb{Z}$. The map to unordered roots is a two-sheeted covering. Its induced subgroup is $2\mathbb{Z}$ inside ${\pi }_1(Y)$, while exchanging roots is a generator; hence ${\pi }_1(Y,p_0)\simeq \mathbb{Z}$.