#
nLab

Betti number

Contents
### Context

#### Homotopy theory

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

#### Homological algebra

# 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.

Last revised on September 14, 2018 at 20:04:12.
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