nLab
Betti number

Context

Homotopy theory

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

For nn \in \mathbb{Z}, the nn-Betti number of a chain complex VV (of modules over a ring RR) is the rank

b n(V):=rk RH n(V) b_n(V) := rk_R H_n(V)

of its nnth homology group, regarded as an RR-module.

For XX a topological space, its nn-Betti number is that of its singular homology-complex

b n(V)=rk RH 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.

Revised on January 6, 2013 20:38:02 by Ingo Blechschmidt (93.104.18.247)