Whether genus can be nonnegative?

Question

From this question, I know the Euler characteristic of $S^2 \times S^2$ is $4$, and I know the Euler characteristic is equal to $2-2g$, where $g$ is the genus. So I can get the genus is $-1$, but I thought the genus must be nonnegative ?!

Answer 1

The Euler characteristic of a closed orientable surface is $2 - 2g$ where $g$ is the genus. The genus is always non-negative.

Note that $S^2\times S^2$ is a closed orientable four-dimensional manifold, not a surface, so the Euler characteristic is not given by $2-2g$.