A topologically invariant property of a surface defined as the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it. Roughly speaking, it is the number of holes in a surface.
The genus of a surface, also called the geometric genus, is related to the Euler characteristic . For a orientable surface such as a sphere (genus 0) or torus (genus 1), the relationship is
For a nonorientable surface such as a real projective plane (genus 1) or Klein bottle (genus 2), the relationship is