= 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 $\mathbb C^*$ and contraction of $\mathbb C$ give $\mathbb C^*\times\mathbb C\simeq S^1$. Coordinates $(a,b)\mapsto(a-b,b)$ identify $X\simeq\mathbb C^*\times\mathbb C$, 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 $\pi_1(Y,p_0)\simeq\mathbb Z$.
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