Euler Characteristics in Geometry
Definition
Simple Definition
Let’s assume we are given a shape. Let’s call the number of vertices $V$, the number of edges $E$, and the number of faces $F$. The Euler characteristic $\chi$ of this shape is defined as follows.
$\chi := V - E + F$
Complex Definition1
For a surface $M$ and its region $\mathscr{R}$, the term $\chi(\mathscr{R}) \in \mathbb{Z}$ that satisfies the Gauss-Bonnet theorem is called the Euler characteristic of $\mathscr{R}$.
$$ \sum_{i=1}^{n} \int_{C_{i}}K_{g}ds + \iint_{R} K dA + \sum\theta_{i} = 2\pi \chi(\mathscr{R}) $$
See Also
Euler’s Formula in Graph Theory
The original Euler characteristic is most famous in graph theory, where the Euler polyhedron theorem or Euler formula is a theorem of graph theory that states $\chi = 2$ for connected planar graphs.
Euler Characteristic in Geometry
It is defined as an integer that satisfies the equation of the Gauss-Bonnet theorem.
Euler Characteristic in Algebraic Topology
It is defined as the alternating sum of the Betti numbers of each dimension.
Richard S. Millman and George D. Parker, Elements of Differential Geometry (1977), p189-190 ↩︎