Genus of surface

Informally, the genus the number of holes in the surface.

Suppose $g$ is the genus. Then it can be computed via the Euler characteristic, $\chi$, as follows: $$\chi = 2 − 2g,$$ for closed surfaces. For surfaces with $b$ boundary components, $$\chi = 2 − 2g − b.$$

The genus is a topological invariant. in fact, it's a complete invariant in the sense that, if two orientable closed surfaces have the same genus, then they must be homeomorphic.

Properties. The genus of surface $S$ is

• half the first Betti number of $S$.
• the number of elements of a set of mutually non-intersecting simple closed curves with the property that $S\setminus C$ is a connected planar surface.

Theorem. Any compact orientable surface $S$ without boundary is a connected sum of $g$ tori, where $g$ is the genus of $S$.

One can think of this number as the number of "handles" in the surface.

See also classification of surfaces.