nLab
Betti number

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

homological algebra

(also nonabelian homological algebra)

Introduction

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 nnth 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) \,.

For XX moreover a smooth manifold then by the de Rham theorem this is equivalently the dimenion of the de Rham cohomology groups.

Properties

Euler characteristic

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

Revised on January 4, 2017 17:56:47 by Mateo Carmona? (186.85.202.64)