Contents

# Contents

## 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.

## References

Named after Enrico Betti.

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Last revised on May 26, 2022 at 10:24:37. See the history of this page for a list of all contributions to it.