# The Klein bottle using the Euler caracteristic + orientability caracterisation

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I went off on my own to study the Euler caracteristic + orentability caracterisation of closed surface and I must have gotten lost somewhere, because I do not find that the Klein bottle is homeomorphic to $\mathbb{R}P^2\#\mathbb{R}P^2$ as I should.

I started with the result that for two surfaces M and N, we have

$$\chi(M\#N)=\chi(M)+\chi(N)-2$$

then I proved by induction that

$$\chi(M^{\#_k})=(k+1)\chi(M)-2k$$

which can be solved for k:

$$k=\frac{\chi(M^{\#_k})-\chi(M)}{\chi(M)-2}$$

Now, the Klein bottle is not orientable, so it must be homeomorphic to $(\mathbb{R}P^2)^{\#_k}$ for some k. And since the Euler caracteristic is a topological invariant, it must be that

$$\chi((\mathbb{R}P^2)^{\#_k})=\chi(K)$$

And so, according to the above formula for k, with $\chi(M)=\chi(\mathbb{R}P^2)=1$,

$$k=\frac{\chi(K)-1}{1-2}=1-\chi(K)$$

So all that remains to do is to triangulate K and find its Euler caracteristic. A fundamental polygon for the Klein bottle is provided here:

http://en.wikipedia.org/wiki/Classification_theorems_of_surfaces#Construction_from_polygons

I believe it can be triangulated simply by drawing a side across one of the diagonal. All the vertices are identified, so V=1. There are 3 distinct sides, so E=3. My triangulation has 2 triangles so F=2, thus

$$\chi(K)=1-3+2=0$$

But this means that k=1, which is incorrect. It should be 2.

Where did I go wrong?

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mathwonk
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well your euler characteristic calculation seems correct, since just looking at a picture of a klein bottle (the picture in my head) shows it obviously has the same edges faces and vertices as a torus, hence has the same euler chracteristic, namely zero. so the error is likely somewhere else.

Homework Helper
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Whoah! There's no error at all !

$$\mathbb{R}P^2\#\mathbb{R}P^2 = (\mathbb{R}P^2)^{\#_1}$$

k=1 is correct! :grumpy:

Edit: No! k should be 2 but I made an error in the very begining by misinterpreting the #_k notation. I thought k refered to the number of # symbol, while it refers to the number of spaces you glue together.

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