## Relationship Between the Genus and Euler Characteristic of a Surface

The genus of a surface is the greatest number of distinct continuous non intersecting closed curves which can be drawn on a surface without separating it into distinct regions. For each division of a surface to become a map, the division has to include edges such that all regions are simply connected, so the euler characteristic of a surface is related to its genus.

The genus and euler characteristic for some surfaces are given in the table.

Surface | Genus | Euler Characteristic |

Sphere | 0 | 2 |

Torus | 1 | 0 |

Two Hole Torus | 2 | -2 |

A Torus With n Holes |

In fact, for any surface the genus g is relation to the Euler characteristicvia the equation