Let $f: X \rightarrow Y$ be a nonconstant holomorphic map between compact connected Riemann surfaces. Define the valency of $f$ at a point, and the degree of $f$.

Define the genus of a compact connected Riemann surface $X$ (assuming the existence of a triangulation).

State the Riemann-Hurwitz theorem. Show that a holomorphic non-constant selfmap of a compact Riemann surface of genus $g>1$ is bijective, with holomorphic inverse. Verify that the Riemann surface in $\mathbb{C}^{2}$ described in the equation $w^{4}=z^{4}-1$ is non-singular, and describe its topological type.

[Note: The description can be in the form of a picture or in words. If you apply RiemannHurwitz, explain first how you compactify the surface.]