#
nLab

Betti number

### Context

#### Homotopy theory

## Background

## Variations

## Definitions

## Paths and cylinders

## Homotopy groups

## 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 $n$-Betti number is that of its singular homology-complex

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

## Properties

### Euler characteristic

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