Betti number

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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

The k-th Betti number

bk(X)

of the space X is defined as the rank of the abelian group

Hk(X),

the k-th homology group of X. Equivalently, one can define it as the vector space dimension

of

Hk(X,Q)

since the homology group in this case is a vector space over Q. The universal coefficient theorem, in a very simple case, shows that these definitions are the same.

More generally, given a field F one can define

bk(X,F),

the k-th Betti number with coefficients in F, as the vector space dimension of

Hk(X,F).

Properties

The Betti numbers bk(X) do not take into account any torsion in the homology groups, but they are very useful basic topological invariants. In the most intuitive terms, they allow one count the number of holes of different dimensions. For a circle, the first Betti number is 1. For a general pretzel the first Betti number is twice the number of holes.

In the case of a finite simplicial complex the homology groups Hk(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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi(K)=\sum_{i=0}^\infty(-1)^ib_i(K,F), }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi(K)} denotes Euler characteristic of K and any field F.

For any two spaces X and Y we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{X\times Y}=P_XP_Y}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_X} denotes the Poincaré polynomial of X, i.e. the generating function of the Betti numbers of X:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_X(z)=b_0(X)+b_1(X)z+b_2(X)z^2+...\ ,}

see Künneth theorem.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_k(X)=b_{n-k}(X) \mbox{ for any } k}

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, ... .

In fact, 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.

Betti number in graph theory

In graph theory a Betti number of a graph G with n vertices, m edges and k connected components equals

m - n + k

de:Bettizahl