I've been studying for a topology exam and recently came across this problem which have no idea how to solve.
Let $\Sigma$ be an orientable closed surface of genus $2$. Fix a base point $x_0\in \Sigma$ and an isomorphism $$ \pi_1(\Sigma,x_0)\cong\langle a,b,c,d \ |\ aba^{-1}b^{-1}cdc^{-1}d^{-1}=1\rangle. $$ Since $\Sigma$ is path connected there is a standard procedure for associating to any oriented loop $\gamma\in \Sigma$ (which may or may not pass though $x_0$) a conjugacy class $C_{\gamma}\subseteq\pi_1(\Sigma,x_0)$. This I understand.
Suppose that $\alpha$ and $\beta$ are oriented loops in $\Sigma$ such that $C_{\alpha}$ is represented by the element $ab^2cda^{-1}b^{-1}c^{-1}d^{-1}$ and $C_{\beta}$ is represented by the element $a^2bcda^{-1}b^{-1}c^{-1}d^{-1}$. Show that it is impossible to use homotopies of loops to make $\alpha$ and $\beta$ disjoint.