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 characterisic 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