nLab Betti number

Theorems

Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

Contents

Definition

For $n\in ℤ$, the $n$-Betti number of a chain complex $V$ (of modules over a ring $R$) is the rank

${b}_{n}\left(V\right):={\mathrm{rk}}_{R}{H}_{n}\left(V\right)$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

${b}_{n}\left(V\right)={\mathrm{rk}}_{R}{H}_{n}\left(X,R\right)\phantom{\rule{thinmathspace}{0ex}}.$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)