Consider the curves $ \gamma_i$ and $ \delta$ on the surface of genus $2$ as pictured with orientations on $\gamma_i$. Show that $ \delta=0$ and $ \gamma_1=\gamma_2\ + \gamma_3\ $ in $H_1$.
The part $\delta= 0$ can be proved using the following fact : $\delta$ can be regarded as a map $f:\Bbb S^1 \rightarrow X $ and then $f$ can be extended to $\tilde f:W \rightarrow X $ where $W$ is an orientable surface with boundary $\Bbb S^1 $.
Need help for proving other part.
