If I understood correctly, a sphere with a single hole is a torus, and has genus 1. Its Euler characteristic is 0.
If it has two holes, the Euler characteristic is -2. Three holes, -4.
However, I don't know how to generalize for a sphere with $n$ holes. Although it seems somewhat intuitive given the initial results, I couldn't prove it formally… Can anybody help me on that?
Best Answer
To elaborate on the previous answer, you know that $\chi(M\#N)=\chi(M)+\chi(N)-\chi(S^2)$ where $M\#N$ is the connected sum. Recognizing that a genus $g$ surface is a genus $g-1$ surface together with a torus, the result follows by an easy induction.