Genus of a surface ($g$)

The genus counts the handles of a connected closed orientable surface. Such a surface has Euler characteristic $2-2g$.

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.