Definition
For a given polyhedron $P$, it is the quantity
$$ \mathcal{X}(P)=V-E+F $$where $V, E,F$ stand, respectively, for the number of vertices, edges and faces of $P$.
Given a compact surface $S$ and a polyhedron $P$ topologically equivalent to $S$, we defined
$$ \mathcal{X}(S):=\mathcal{X}(P) $$$\blacksquare$
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.
$\blacksquare$
Key idea: triangulation does not affect Euler characteristic.
It is also related to the Poincare-Hopf Theorem.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: