## Euler Characteristic of a Surface With Handles, Crosscaps and Holes

A sphere with one handle, one cross cap and two holes is shown below.

To detach the handle leaves two holes and the crosscap one hole. On a sphere with n holes, m handles and c crosscaps, n+2m+c holes are left by detaching all the handles and crosscaps. When the handles are detached they become cylinders with Euler characteristic zero.

Hence

A surface expressed as a sphere with n holes, m handles and c crosscaps is said to be in standard form. Some surfaces are represented in standard form in the following table.

Surface | holes | handles | Crosscaps | |

Sphere | 0 | 0 | 0 | 2 |

Disc | 1 | 0 | 0 | 1 |

Cylinder | 2 | 0 | 0 | 0 |

Real Projective Plane | 0 | 0 | 1 | 1 |

Mobius Band | 1 | 0 | 1 | 0 |

Torus | 0 | 1 | 0 | 0 |

Two Hole Torus | 0 | 2 | 0 | -2 |

n Hole Torus | 0 | n | 0 | 2-2n |