Arbitrary-degree maps to a surface of genus two
= Arbitrary-degree maps to a surface of genus two
{title2=$\Sigma_{|k|+1}\longrightarrow\Sigma_2,\quad\deg=k$}
For $k>0$, an epimorphism $\pi_1(\Sigma_2)\to\mathbb Z/k$ gives a connected $k$-sheeted cover. Its Euler characteristic is $-2k$, hence its genus is $k+1$, and the lifted orientation gives degree $k$. For negative $k$ compose the $|k|$-sheeted cover with an orientation-reversing homeomorphism of the target. For $k=0$ take a constant map from the torus.