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

$$b_n(V):={\mathrm{rk}}_R{H}_n(V)$$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 $n$-Betti number is that of its singular homology-complex

$$b_n(V)={\mathrm{rk}}_R{H}_n(X,R).$$b_n(V) = rk_R H_n(X, R) \,.

Properties

Euler characteristic

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