Surface Classification
Every compact connected surface is homeomorphic to one of the following:
- a sphere
- a connected sum of n tori (e.g., simple n-holed donut)
- a connected sum of n projective planes for non-orientable case
The Euler characteristics (X)
(surface integral of Gaussian curvature)
of these surfaces is given by:
- a sphere has Euler characteristic 2
- n_tori has Euler characteristic 2-2n
- m_projective planes has Euler characteristic 2-m
Thus orientable surfaces can only have even Euler characteristic
whereas non-orientable surfaces can have Euler characteristic
equal to any number less than or equal to 1.
E.g., the Euler characteristic of the Klein bottle is 0.
The genus of a surface (max number of independent closed cuts that leave the surface in one piece)
is defined as follows (where X is the Euler characteristic):
- for orientable surfaces : G = 1-X/2
- for nonorientable surfaces : G = 2-X
So the Klein bottle is non-orientable of genus 2.
This info can be found in an easy-to-read undergraduate text
"Topology of Surfaces" by L Kinsey.