I have a question about the fondamental group of the following covering space
$$ p : Y \rightarrow X ; \; Y \owns (x,y) \mapsto x \in X $$ where $X = {\mathbb P}^1$ and
$$ Y := \{(x,y) \in {\mathbb C}^2 : y^2 = (x-a_1)(x-a_2)(x-a_3)(x-a_4)\}. $$
Denote a closed path on $X$ based at $x_0$ encircling $a_i$ once by $\gamma_i$. Then we have
$$ \pi_1(X,x_0) = \{[\gamma_1],[\gamma_2],[\gamma_3],[\gamma_4] : [\gamma_1][\gamma_2][\gamma_3][\gamma_4]=1\}, $$ where $x_0 \in X$ and $[\gamma_i]$ is the homotopy class of $\gamma_i$.
We identify $Y$ with a torus by cutting the surface of $X$ along lines from $a_1$ to $a_2$ and $a_3$ to $a_4$ and pasting two copies of them along the lines.
We denote the meridian circle and the latitude cicle of the torus by $\alpha$ and $\beta$ respectively. Then,
$$ \pi_1(Y,y_0) = \{[\alpha],[\beta]\ \; : \; [\alpha][\beta][\alpha]^{-1}[\beta]^{-1}=1 \}, $$ where $y_0 \in p^{-1}(x_0)$. Then, $$ p_*([\alpha]) = [\gamma_1][\gamma_2], \; p_*([\beta]) = [\gamma_2][\gamma_3]. $$
Combining the condition $[\gamma_1][\gamma_2][\gamma_3][\gamma_4]=1$, we have
$$ \gamma_3 \equiv \gamma_1 {\rm mod} \; p_* \pi_1(Y,y_0), \; \gamma_2 \equiv \gamma_4 \equiv \gamma_1^{-1} {\rm mod} \; p_* \pi_1(Y,y_0) $$
But I cannot derive $$ \gamma_1^2 \equiv 1 \; {\rm mod} \; p_* \pi_1(Y,y_0), $$ which I think necessary to complete the conditions.
Could someone help me?