Euler-Poincaré Theorem

Recall that the order of a group $G$ is simply its cardinality $|G|$. We define the rank of a group $G$ as the smallest generating set, i.e.,

$$\text{rank}G=\min \{|X|:X\subseteq G,⟨X⟩=G\}.$$

When working over ${\mathbb{Z}}_2$, there are $2^n$ $p$-chains where $n$ is the number of $p$-simplices in $K$. We can generate them using the elementary $p$-chains $e_1,\dots ,e_n$. Thus, one may see that $\text{rank}{C}_p=n$.

Assume that $\{b_1,\dots ,b_k\}$ is the smallest generating set for $B_p$. Since $B_p\le Z_p$, we can extend to a generating set $\{b_1,\dots ,b_k,z_1,\dots ,z_j\}$ for $Z_p$. So

$$H_p=\{z+B_p:z\in Z_p\}=⟨z_1+B_p,\dots ,z_j+B_p⟩.$$

Hence,

$$\text{rank}{H}_p=\text{rank}{Z}_p-\text{rank}{B}_p.$$

The rank of $H_p$ is called the p-th Betti number.

Betti number

The p-th Betti number ${\beta }_p$ is given by

$${\beta }_p=\text{rank}{H}_p=\text{rank}{Z}_p-\text{rank}{B}_p$$

where $H_p$ is the p-th homology group.

For $p=0,1,$ and 2 the Betti number’s give an intuitive implication:

• ${\beta }_0$ is the number of connected components
• ${\beta }_1$ is the number of holes
• ${\beta }_2$ is the number of enclosed spaces.

As a result, the Betti numbers capture the underlying topology of the complex.

Recall the Euler characteristic:

$$\chi (K)=\sum _{p=0}^{\dim K}(-1{)}^p{s}_p$$

where $s_p$ is the number of $p$-simplices in $K$. Notice that $s_p=\text{rank}{C}_p$. Let $z_p=\text{rank}{Z}_p$ and $b_p=\text{rank}{B}_p$.

If we consider the boundary map ${\partial }_p:C_p\to C_{p-1}$ as a linear transformation, then it follows from rank-nullity that

$$\dim C_p=\text{nullity}{\partial }_p+\text{rank}{\partial }_p$$

Therefore, we can see that

$$s_p=z_p+b_{p-1}.$$

Applying this equality to the Euler characteristic gives us the Euler-Poincaré theorem:

Euler-Poincaré

$$\chi (K)=\sum _{p=0}^{\dim K}(-1{)}^p{\beta }_p$$

As a consequence, we can see that the Homology groups are independent of the chosen triangulation.