Betti number

In algebraic topology, the Betti number of a topological space is, in intuitive terms, a way of counting the maximum number of cuts that can be made without dividing the space into two pieces. This defines, in fact, what is called the "first" Betti number. There is a sequence of Betti numbers defined.

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, the name being for Enrico Betti.

Definition

For a non-negative integer $k$, the $k$-th Betti number $b\_\{k\}\; (X)$ of the 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 the vector space dimension of $H\_\{k\}\; (X,\; mathbb\; Q)$, since the homology group in this case is a vector space over $mathbb\; Q$. 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 $b\_\{k\}(X,F)$, the $k$-th Betti number with coefficients in $F$, as the vector space dimension of $H\_\{k\}(X,F)$.

Example: the first Betti number in graph theory

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

:"m" − "n" + "k".

This may be proved straightforwardly by mathematical induction on the number of edges. A new edge either increments the number of 1-cycles or decrements the number of connected components.

See cyclomatic complexity for an application of the first Betti number in software engineering.

Properties

The (rational) 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 most intuitive terms, they allow one to 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 $H\_\{k\}\; (X,\; mathbb\; Z)$ 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:$chi(K)=sum\_\{i=0\}^infty(-1)^ib\_i(K,F)\; ,\; ,!$where $chi(K)$ denotes Euler characteristic of "K" and any field "F".

For any two spaces "X" and "Y" we have:$P\_\{X\; imes\; Y\}=P\_XP\_Y\; ,\; ,!$where $P\_X$ denotes the Poincaré polynomial of "X", i.e. the
generating function of the Betti numbers of "X"::$P\_X(z)=b\_0(X)+b\_1(X)z+b\_2(X)z^2+cdots\; ,\; ,!$see Künneth theorem.

If "X" is "n"-dimensional manifold, there is symmetry interchanging "k" and "n" − "k", for any "k"::$b\_k(X)=b\_\{n-k\}(X)\; ,\; ,!$under conditions (a "closed" and "oriented" manifold); see Poincaré duality.

The dependence on the field "F" is only through its characteristic. If the homology groups are torsion-free, the Betti numbers are independent of "F". The connection of "p"-torsion and the Betti number for characteristic p, for "p" a prime number, is given in detail by the universal coefficient theorem (based on Tor functors, but in a simple case).

Examples

#The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...;
#The Betti number sequence for a two-torus is 1, 2, 1, 0, 0, 0, ...;
#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. This is a case of the Künneth theorem.

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 when $X$ is a closed manifold, 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.

In this setting, Morse theory gives a set of inequalities for alternating sums of Betti numbers in terms of a corresponding alternating sum of the number of critical points $N\_i$ of a Morse function of a given index:

:$b\_i(X)\; -\; b\_\{i-1\}\; (X)\; +\; cdots\; ge\; N\; \_i\; -\; N\_\{i-1\}\; +\; cdots$

Witten gave an explanation of these inequalities by using the Morse function to modify the exterior derivative in the de Rham complex.

References

* F.W. Warner, Foundations of differentiable manifolds and Lie groups, Springer (1983).
* J.Roe, Elliptic Operators, Topology, and Asymptotic Methods, Second Edition (Research Notes in Mathematics Series 395), Chapman and Hall (1998).