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.