Betti number - Definition and Overview

In algebraic topology, the Betti numbers of a topological space X are a sequence b₀, b₁, ... of topological invariants. Each Betti number is a natural number, or infinity. For the most reasonable spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some points onwards, and consists of natural numbers. The term "Betti numbers" was coined by Henri Poincaré and named for Enrico Betti.

Definition

k-th Betti number b_k(X) of space X is defined as the rank of the abelian group H_k(X), the k-th homology group of X. Equivalently one can define it as dimension of H_k(X,Q) since the homology group in this case is a vector space over Q.

More generally, given a field F one can define b_k(X,F) (the k-th Betti number with coefficients in F) as the dimension of H_k(X,F).

Properties

The Betti numbers b_k(X) do not take into account any torsion in the homology groups, but they are very useful basic topological invariants.

In the case of a finite simplicial complex the homology groups H_k(X,Z) are finitely-generated, and so has a finite rank. Also the group is 0 when k exceeds the top dimension of a simplex of X.

For a finite CW-complex K we have

<math>\chi(K)=\sum_{i=0}^\infty(-1)^ib_i(K,F), <math>

where <math>\chi(K)<math> denotes Euler characteristic of K and any field F.

For any two spaces X and Y we have

<math>P_{X\times Y}=P_XP_Y<math>

where <math>P_X<math> denotes the Poincaré polynomial of X, i.e. the generating function of the Betti numbers of X:

<math>P_X(z)=b_0(X)+b_1(X)z+b_2(X)z^2+...\ ,<math>

see Künneth theorem.

If X is n-dimensional manifold, there is symmetry interchanging k and n − k:

<math>b_k(X)=b_{n-k}(X) \mbox{ for any } k<math>

see Poincaré duality.

Examples

1. The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...;
2. The Betti number sequence for a two-torus is 1, 2, 1, 0, 0, 0, ...;
3. The Betti number sequence for a three-torus is 1, 3, 3, 1, 0, 0, 0, ... .

Infact, for an n-torus one should indeed see the binomial coefficients.

It is possible for spaces that are infinite-dimensional in an essential way to have an infinite sequence of non-zero Betti numbers. An example is the infinite dimensional complex projective space, with sequence 1, 0, 1, 0, 1, ... that is periodic, with period length 2.

Relationship with dimensions of spaces of differential forms

In geometric situations, the importance of the Betti numbers may arise from a different direction, namely that they predict the dimensions of vector spaces of closed differential forms modulo exact differential forms. The connection with the definition given above is via three basic results, de Rham's theorem and Poincaré duality (when those apply), and the universal coefficient theorem of homology theory.

There is an alternate reading, namely that the Betti numbers give the dimensions of spaces of harmonic forms. This requires also the use of some of the results of Hodge theory, about the Hodge Laplacian.