I want to calculate the homology of a surface of genus $g$ using the Mayer-Vietoris Sequence (MVS).
In this post (Computing the homology of genus $g$ surface, using Mayer-Vietoris) it is addressed how one can reason that the map $(j_U)_*$ on homology $H_1(U\cap V) \rightarrow H_1(U)\oplus H_1(V) \cong H_1(U)$ induced by the inclusion $j_U$ is trivial. I do not understand why this is the case. The rest of the computation is clear to me.
Can I see this in a picture? I am having trouble understanding what an induced map on homology does to a particular cycle.