Euler-Poincare characteristic

Definition

• For a polyhedron:
  For a given polyhedron $P$, it is the quantity

where $V,E,F$ stand, respectively, for the number of vertices, edges and faces of $P$.

• For a compact surface:
  Given a compact surface $S$ and a polyhedron $P$ topologically equivalent to $S$, we defined

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Proposition (Euler Fórmula)
For a surface $S$, Euler characteristic satisfies $\mathcal{X}(S)=2-2g$, being $g$ the genus of the surface.

Proof
Step 1. If a polyhedron is topologically equivalent to a sphere the $\mathcal{X}(P)=2.$ Therefore $\mathcal{X}({\mathbb{S}}^2)=2-2\cdot 0=2$ and $\mathcal{X}(S)=2$ for any $S$ of genus equals to 0.
See Needham_2021 page 186 for Cauchy's proof.

Step 2. If we attach a handle to a compact surface $S$, we reduces $\mathcal{X}(S)$ by 2. See Needham_2021 page 192.

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Key idea: triangulation does not affect Euler characteristic.

It is also related to the Poincare-Hopf Theorem.