---
tag: V27468ECYW
kind: definition
name: Genus
msc: [57K20]
defines: NJ787GZS1N
uses: [VGJJVBZ1MQ, 0TD904MG06, 0V032VWTHD]
sources:
  - url: https://en.wikipedia.org/wiki/Genus_(mathematics)
---

## Definition

Let $S$ be a closed [[orientable|0TD904MG06]] [[surface|VGJJVBZ1MQ]], that is, a compact connected orientable surface without boundary.
Call simple closed curves $c_1, \ldots, c_k$ in $S$ *non-separating* when they are pairwise disjoint and $S \setminus (c_1 \cup \cdots \cup c_k)$ is [[connected|0V032VWTHD]].
The *genus* $g(S)$ is the largest integer $k \ge 0$ for which such curves $c_1, \ldots, c_k$ exist.

## Remarks

The genus is finite and satisfies $\chi(S) = 2 - 2g(S)$, where $\chi$ is the Euler characteristic.
By the classification of surfaces, $S$ is homeomorphic to the sphere if $g = 0$, and to the connected sum of $g$ tori if $g \ge 1$.

Example: for the torus $T^2 = S^1 \times S^1$, the curve $c_1 = S^1 \times \set{1}$ has connected complement $S^1 \times (S^1 \setminus \set{1})$, an open annulus.
Any two disjoint simple closed curves in the torus disconnect it, so $g(T^2) = 1$, in accord with $\chi(T^2) = 0$.
Non-example of a non-separating curve: on the sphere $S^2$ every simple closed curve separates the sphere into two pieces, by the Jordan curve theorem, so $g(S^2) = 0$.

For a closed non-orientable surface, the *non-orientable genus* is the number $k$ of copies of $\R P^2$ in its connected sum decomposition; then $\chi = 2 - k$.

## Explanations

### L1 human

The genus counts how many separate cuts along closed loops a surface can survive while staying in one piece.
Each handle allows exactly one such cut, around the handle.
So the genus is the number of handles.