I am trying to solve the following exercise:
Let $M_4$ be the orientable closed surface of genus $4$ and $\alpha_i,\alpha_i',\beta_i, \gamma_i$ the curves given as in the following figure:
Find a basis of $H_1(M_4)$ in ther of the cycles defined by the curves $\alpha_i,\alpha_i',\beta_i, \gamma_i$.
For do so, I use the fact that $H_1(M_4)$ is isomorphic to the abelianized $\pi_1(M_4)_{ab}$ of the fundamental group $\pi_1(M_4)$. This fundamental group could be given as a presentation group in terms of the homotopy classes of loops $a_1,b_1,\ldots,a_4,b_4$ at some point $x_0$ which are defined by the 1-cells of $M_4$: $$\pi_1(M_4)\cong \langle a_1,b_1,\ldots, a_4,b_4\rangle|[a_1,b_1]\dots[a_4,b_4]\rangle.$$ Thus $H_1(M_4)\cong \langle a_1,b_1,\ldots, a_4,b_4\rangle$. From here, I would like to show that each curve $\beta_i$ is homotopic to $b_i$ and $\gamma_i$ to $a_1$. Then trivially $H_1(M_4)$ can be written in terms of $\{\beta_i,\gamma_i\}$.
Any help would be appreciated.