Describe the Riemann surface for $w=z^2-1$.

Question

Question:

Describe the Riemann surface for $w=z^2-1$.

My thoughts so far: the Riemann surface needs two cuts emanating from the origin across the real line since it maps every input to one point above the real line and one point below. This means we would need two "sheets" to make the mapping one-to-one.

Answer 1

Here is what should be the correct answer. We have $z=\sqrt{w+1}$ with a single branch point at $-1$. So, following the construction of the surface for $z^2=w$, we cut two copies of the plane along the real axis from $-1$ to infinity and glue them together along the slit.

The compactification should have genus $0$ (since all irreducible conics are topologically equivalent to the Riemann sphere).