Course MT4521 Geometry and topology

## The Euler Characteristic

The *Euler Characteristic* is something which generalises Euler's observation of 1751 (in fact already noted by Descartes in 1639) that on "triangulating" a sphere into *F* regions, *E* edges and *V* vertices one has *V* - *E* + *F* = 2.

If one triangulates *any* surface then *χ* = *V* - *E* + *F* is a number which does not depend on how the triangulation is done. This is called the **Euler Characteristic**.

**Examples**

*χ*(Torus) = 0

For example we can split up the torus into "regions" (homeomorphic to discs) with *F* = 2, *E* = 4, *V* = 2 or even more efficiently as with *F* = 1, *E* = 2, *V* = 1

*χ*(Projective plane) = 1

The planar model with edge word *a*^{2} gives a triangulation with *F* = 1, *E* = 1, *V* = 1

*χ*(Klein bottle) = 0

The model gives a triangulation with *F* = 1, *E* = 2, *V* = 1

**Theorem**

*χ*(*S*_{1} # *S*_{2}) = *χ*(*S*_{1}) + *χ*(*S*_{2}) - 2

**Proof**

Join *S*_{1} and *S*_{2} by removing an *n*-gon from each and matching vertices around the edge.

Then: *V*(*S*_{1} # *S*_{2}) = *V*(*S*_{1}) + *V*(*S*_{2}) - *n*,

*E*(*S*_{1} # *S*_{2}) = *E*(*S*_{1}) + *E*(*S*_{2}) - *n*,

*F*(*S*_{1} # *S*_{2}) = *F*(*S*_{1}) + *F*(*S*_{2}) - 2

and so *χ*(*S*_{1} # *S*_{2}) = *χ*(*S*_{1}) + *χ*(*S*_{2}) - 2.
**Corollary**

For the join of *n* tori we have *χ* = -2(*n* - 1) and for the join of *n* Projective planes we have *χ* = -(*n* - 2).

**Proof**

Use induction and *χ*(*T*) = 0, *χ*(*P*) = 1.

So if we could recognise whether a surface could be written as a product of tori or of projective planes, the Euler characteristic would be enough to classify it.

**Definition**

If when we carry a direction around any loop in a surface we always get a consistent direction the surface is called **orientable**.

**Examples**

- A sphere and any join of tori are orientable.

**Proof**

The direction of an "outward-pointing normal" can be used to define the direction.

- A Möbius band is not orientable.

**Proof**

Carry a direction around the curve "in the middle of the band". (A normal at a point would come back pointing the opposite way after going round once.)

- A Projective plane or any join of projective planes is not orientable.

**Proof**

These all contain a Möbius band.

- Any surface whose planar model contains an edge which is identified with itself rather than its inverse is not orientable.

**Proof**

Cut a strip out of the model with ends on the identified edge and you have a Möbius band.

JOC February 2010