Genus

Summary

The genus of a closed orientable surface is its number of handles. It is the largest number of disjoint simple closed curves along which the surface can be cut without falling apart. By the classification of surfaces, the genus determines a closed orientable surface up to homeomorphism, and it is related to the Euler characteristic by $\chi = 2 - 2g$. The same number appears as the genus of an algebraic curve and of a compact Riemann surface.