Euler characteristic

Author: Leandro Alegsa Creation date: November 13, 2022

In mathematics, the Euler characteristic of a shape is a number that describes a topological space, so that anything in the space will have the same number. It is calculated by taking the number of points in the shape, the number of lines in the shape, and the number of faces of the shape. One can find the Euler characteristic with this formula:

\chi =V-E+F\,\!

where V is the point count, E the line count, and F the face count. For most common shapes (convex polyhedron), the Euler characteristic is 2.

For spheres, the Euler characteristic is also 2.

Definition

For surfaces

A closed surface can always be triangulated, that is, one can always cover it with a finite triangular lattice. The Euler characteristic χ $\chi$ is then defined as

$\chi (S):=E-K+F.$

where is the number of vertices, is the number of edges, and is the number of triangles in the triangulation.

For CW complexes

Let be a topological space that is a finite-dimensional CW-complex Let k_i denote the number of cells of dimension and let be the dimension of the CW-complex. Then the Euler characteristic is given by the alternating sum

$\chi (X):=\chi (T)=\sum _{{i=0}}^{n}(-1)^{i}k_{i}$

is defined. This Euler characteristic for CW-complexes is also called Euler-Poincaré characteristic. If one decomposes the space into simplices instead of cells, one can also define the Euler characteristic analogously by the resulting simplicial complex . For the Euler characteristic holds

$\chi (X):=\chi (C)=\sum _{{i=0}}^{{n}}(-1)^{{i}}f_{{i}}$

where $f_{{i}}$ is the number of -dimensional simplices of . For a simplicial complex of a two-dimensional space, we obtain $E=f_{{0}}$ , $K=f_{{1}}$ and $F=f_{{2}}$ , we recover the definition of the Euler characteristic on surfaces. The value of the characteristic is independent of the type of calculation.

Definition by means of singular homology

Let again be a topological space. The rank of the -th singular homology groups is called the -th Betti number and is $b_{i}$ denoted by If the singular homology groups have finite rank and only finitely many Betti numbers are nonzero, then the Euler characteristic of given by

$\chi (X):=\sum _{{i=0}}^{n}(-1)^{i}b_{i}=\sum _{{i=0}}^{n}(-1)^{i}\dim(H_{i}(X))$

defined. If is a CW-complex, then this definition gives the same value as in the definition for CW-complexes. For example, a closed orientable differentiable manifold satisfies the conditions on singular homology.

Properties

Well-defined

An important observation is that the given definition is independent of the triangular lattice chosen. This can be shown by moving to a joint refinement of given lattices without changing the Euler characteristic.

Moreover, since homeomorphisms preserve a triangulation, the Euler characteristic even depends only on the topological type. Conversely, if two surfaces have different Euler characteristics, it follows that they must be topologically different. Therefore it is called a topological invariant.

Relationship to the sex of the area

The Euler characteristic χ $\chi$ and the gender of the surface are related. If the surface is orientable, then the relation holds.

$\chi (S)=2-2g,$

If the surface is not orientable, on the other hand, the equation

$\chi (S)=2-g.$

This formula for orientable surfaces results as follows: We start with a 2-sphere, i.e., a surface of gender 0 and Euler characteristic 2. A surface of gender obtained from it by -folding the connected sum with a torus. The connected sum can be set up so that the gluing occurs along one triangle of the triangulation at a time. This gives the following balance per gluing:

Surfaces: (the two bonding surfaces).
Edges: (each 3 edges are glued, they then count only once).
Corners: (each 3 corners are glued, they also count only once).

so in total χ $\chi '=\chi -3+3-2=\chi -2$ . Thus, by each of the tori, the Euler characteristic decreases by 2.

Connection with the Eulerian polyhedron theorem

Let be a convex polyhedron that can be embedded in the interior of a 2-sphere $\mathbb{S}^2$ can be embedded. Now one can consider the vertices, edges and exterior faces of this polyhedron as cells of a CW-complex. Also the singular homology groups of the complex are finite dimensional. Since the polyhedron is orientable and has gender 0, it follows from the above section that the Euler characteristic has value 2. Altogether we get the formula

E-K+F=2 ,

where describes the number of vertices, the number of edges and the number of faces. This formula is called Euler's polyhedron formula.