Genus monotonicity under nonzero-degree surface maps ($\mathrm{deg}f\ne 0⟹g(M)\ge g(N)$)

For a continuous map of nonzero mapping degree between closed connected oriented surfaces, rational cohomology injectivity of a nonzero-degree map gives an injection on $H^1$. Since the first Betti number of an oriented surface of genus $g$ is $2g$, the source genus is at least the target genus.