nLab
Betti number

Definition

For $n \in \mathbb{Z}$, the $n$-Betti number of a chain complex $V$ (of modules over a ring $R$) is the rank

$$b_n(V) := rk_R H_n(V)$$

of its $n$th homology group, regarded as an $R$-module.

For $X$ a topological space, its $n$th Betti number is that of its singular homology-complex

$$b_n(V) = rk_R H_n(X, R) \,.$$

For $X$ moreover a smooth manifold then by the de Rham theorem this is equivalently the dimension of the de Rham cohomology groups.

Properties

Euler characteristic

The alternating sum of all the Betti numbers is – if it exists – the Euler characteristic.

Related concepts

• Hodge number

• etale topos