Euler Characteristic & Surface Classification

Genus, orientability, and = V - E + F classifying closed surfaces.

The idea

The Euler characteristic is a single integer, computed by counting, that tells surfaces apart. Cut a surface into polygons and set $\chi = V - E + F,$ vertices minus edges plus faces. A cube has $V = 8$, $E = 12$, $F = 6$; a tetrahedron has $4$, $6$, $4$. Both are decompositions of the sphere, and both give $\chi = 2$ — as does every other way of cutting the sphere into polygons. The number measures the surface, not the cutting.

To see why, examine one move at a time. Draw a new edge across a face: $F$ rises by one, but so does $E$, and the alternating sum does not change. Place a new vertex on an edge: $V$ rises by one, the edge splits into two, and again nothing changes. Every refinement of a decomposition is assembled from such moves, so $\chi$ cannot depend on which decomposition produced it.

This makes $\chi$ a topological invariant: two surfaces with different values of $\chi$ cannot be homeomorphic. For closed surfaces it is sharper still. A closed orientable surface is a sphere with some number of handles attached — that number is its genus $g$ — and $\chi = 2 - 2g$, so the genus and the Euler characteristic determine each other and one integer, obtained by counting, carries the whole classification.

Ways to work on it

• Walkthrough. Compute = V - E + F and read off genus via = 2 - 2g.
• Practice. Recover the genus of an orientable surface from its Euler characteristic.
• Hardest. Classify non-orientable surfaces from Euler characteristic and orientability.

Not sure where to start? Take the ten-question placement test.