Betti number

(also nonabelian homological algebra)

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 dimenion of the de Rham cohomology groups.

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

Revised on January 4, 2017 17:56:47
by Mateo Carmona?
(186.85.202.64)