Theory

Betti numbers, Euler Characteristic

Rank of an abelian group

Every finitely generated abelian group $G$ is isomorphic to a group with this form:

$${\mathbb{Z}}^r⨁({\oplus }_i\mathbb{Z}/k_i\mathbb{Z})$$

• The ${\mathbb{Z}}^r$ summands give a free abelian part and contain all infinite-order generators.
• The $\mathbb{Z}/k_i\mathbb{Z}$ summands give torsion and contain all finite-order generators.

The rank of $G$ is the number $r$.

Betti numbers

The Betti numbers ${\beta }_i$ of a topological space $X$ are the ranks of the homology groups:

$${\beta }_i(X)=\mathrm{rank}(H_i(X;\mathbb{Z}))$$

These Betti numbers may be combined into a single number, the Euler characteristic:

Euler characteristic

The Euler characteristic $\chi (X)$ of a topological space $X$ is the alternating sum of Betti numbers:

$$\chi (X)=\sum _i(-1{)}^i{\beta }_i(X)$$

Illustration - Euler characteristic of polyhedra and simplicial complexes

For convex polyhedra, Euler’s formula for the Euler characteristic is simply vertices minus edges plus faces:

$$\chi =V-E+F$$

For nonconvex polyhedra, this formula can give values other than the actual Euler characteristic defined using Betti numbers.

Name	Figure	Vertices	Edges	Faces	$\chi =V-E+F$
Tetrahedron		4	6	4	2
Cube		8	12	6	2
Octahedron		6	12	8	2
Dodecahedron		20	30	12	2
Icosahedron		12	30	20	2
Tetrahemihexahedron		6	12	7	1
Cubohemioctahedron		12	24	10	-2
Small stellated dodecahedron		12	30	12	-6

For a simplicial complex, the Euler characteristic can always be computed using a formula analogous to the Euler formula: count the simplices of each dimension:

$$\chi =k_0-k_1+k_2-k_3+\cdots \pm k_n$$

Here $k_i$ is the number of $i$-simplices in the simplicial complex.

Theorem: Euler $=$ Euler

For a simplicial complex, the same Euler characteristic may be calculated using Betti numbers or using the counts of simplices.

We do not prove this here, although the proof is a straightforward homology calculation.