@0celo7 I have le construction. Say you have two disjoint loops $\gamma_1 \cup \gamma_2$ (oriented counterclockwise) representing the class $\alpha \in H_1(M)$. Introduce a path $\sigma$ from $p \in \gamma_1$ to $q \in \gamma_2$ and take a strip neighborhood $U$ around $\sigma$ which hits $\gamma_i$ on little arcs $A_1$ and $A_2$ around $p$ and $q$.
Let $\sigma_1$ and $\sigma_2$ be two paths with endpoints in $A_1$ and $A_2$, lying inside $U$, such that they intersect (so they look like an "X") and orient them so $\gamma_1 \cup \sigma_1 \cup \gamma_2 \cup \sigma_2$ is an immersed loop inter…