# Mismatching Euler characteristic of the Torus

Why is it that when I try to compute the Euler characteristic for the Torus using a drawing like the following , then the number that I get is not the number that the Torus should have? Which is $$0$$?

I mean, if I understand correctly, given any polygon and counting $$V-E+F$$ then I get the Euler characteristic for that specific volume. But in my case, $$V-E+F=16-24+10=2$$.
Obviously, this torus is not the same as the sphere which has the same apparent Euler characteristic.

I know that there exists a way for creating a torus from a simple sheet of paper by identifying the corners of such, and in this case the Euler characteristic comes up being $$0$$ as it should.
But I just wanted to know what was the flaw in my argument.

• "Given any polygon" - you have to define what a "polygon" means correctly. In your current drawing you are not using the right notion of a "polygon." (What you are counting as a top/bottom "face" is not a face by any standard topological definition.) Commented 2 days ago
• Summary : "what was the flaw in my argument" : You can not have holes in the 2 faces, How to fix it ? You have to break the 2 "hole faces" into 2+2 faces without holes. It will change $F$. Naturally , that will change $E$ too , though $V$ will not change.
– Prem
Commented 2 days ago
• Maybe #faces is 8. Commented 2 days ago
• Actually , #faces should be 12 , @BobDobbs , Details in my answer.
– Prem
Commented 2 days ago
• Yes , that is necessary , @BobDobbs , though it is not necessary to add vertices.
– Prem
Commented 2 days ago

## 2 Answers

Euler Characteristic demands no holes in the faces (or rather , the formula changes when there are holes)

Here is a way to remove the holes on the top face and bottom face.

We join 2 Pairs of existing vertices (Purple lines) to generate a new face in green on the top.
We like-wise join 2 Pairs of existing vertices (Blue lines) to generate a new face in grey on the bottom.
Now Euler Characteristic will work out.
$$V-E+F=16-28+12=0$$

[[
Copy-Pasting the gist of my earlier comment here , for quick reference :
Summary :
"what was the flaw in my argument" : You can not have holes in the $$2$$ faces.
How to fix it ? You have to break the $$2$$ "hole faces" into $$2+2$$ faces without holes. That will change $$F$$ , which will naturally change $$E$$ too , though $$V$$ will not change.
]]

You should use Euler formula on a triangulation if you want to compute the euler characteristic. One easy triangulation of the torus can be obtained as following:

Obtained by "discretizing a donut".
Opening up the diagram one obtains (sorry for the drawing)

from which you easily deduce that this particular triangulation has $$9$$ vertex, $$27$$ edges and $$18$$ faces.

• One does not need a triangulation, a CW complex structure would work just fine, this is what OP was trying to use (incorrectly). Commented 2 days ago