Classify this compact surface:
where the edges with the same letter identify.
Using the clasification theorem of compact surfaces I get that this is a proyective plane:
The symbol is abcdabcd wich, after appliying the symbol transforation rules is equivalent to aa , the proyective plane
But if try to classify it using the Euler characteristic I get that χ(X) = $0$ and as I know that this surface is orientable (the 'polygon' does not contain any Mobius strip, anyway I am not really sure about this) then it would be a sphere.
I don´t know what I am doing wrong. Any help is apreciated!