Euler characteristic of the projective plane (using embedding diagram)

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You can see four squares in the figure. The two diagonally opposite squares form a face, as you can see from the gluing. That is why it has two faces in total.

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thinker
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Updated on August 01, 2022

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  • thinker
    thinker over 1 year

    Make the square into the projective plane $\mathbb{P}$ by identifying edges and compute the Euler characteristic by embedding the following graph onto the surface:

    graph

    Here is my diagram of the identified square of the projective plane with the embedded graph on it:

    Projective plane

    Now I can see that $v=3$ and $e=4$ but I am unsure how to calculate the number of faces from this. According to my lecturer the answer is $2$ leading to an euler characteristic of $\chi(\mathbb{P})=v-e+f=3-4+2=1$.

    Could you explain why $f=2$ in this case?

    • Riccardo
      Riccardo over 7 years
      Oh man, just read something else:) my bad!
    • Gerard
      Gerard over 6 years
      e=4 because: A) there are 4 coloured line pieces? Or B) there are 2 couloured lines + 2 border lines?
  • thinker
    thinker over 7 years
    thank you. I suppose for torus and klein bottle will have $f=1$ since the space 'wraps around' itself
  • Henry
    Henry over 7 years
    @thinker exactly!