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Betti number
 
 
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In algebraic topology
Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant that classification theorem topological spaces up to homeomorphism....
, the Betti number of a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
 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
Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
, or infinity
Infinity

Infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology....
. For the most reasonable finite-dimensional spaces (such as compact manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
s, finite simplicial complex
Simplicial complex

In mathematics, a simplicial complex is a topological space of a particular kind, constructed by "gluing together" Point s, line segments, triangles, and their n-dimensional counterparts ....
es or CW complex
CW complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. The idea was to have a class of spaces that was broader than simplicial complexes , but still retained a combinatorial nature, so that computational considerations were not ignored....
es), the sequence of Betti numbers is 0 from some points onwards (Betti numbers vanish about the dimension of a space), and consists of natural numbers (i.e., finite, not infinity).

The term "Betti numbers" was coined by Henri Poincaré
Henri Poincaré

Jules Henri Poincar? was a French mathematician and theoretical physicist, and a philosophy of science. Poincar? is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime....
, the name being for Enrico Betti
Enrico Betti

Enrico Betti was an Italian mathematician, now remembered mostly for his 1871 paper on topology that led to the later naming after him of the Betti numbers....
.

Definition
For a non-negative integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
 , the -th Betti number of the space is defined as the rank
Rank of an abelian group

In mathematics, the rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space over the rational numbers one would need to "contain" it; or alternatively how large a free abelian group it can contain as a subgroup....
 of the abelian group
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
 , the -th homology group of .






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In algebraic topology
Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant that classification theorem topological spaces up to homeomorphism....
, the Betti number of a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
 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
Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
, or infinity
Infinity

Infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology....
. For the most reasonable finite-dimensional spaces (such as compact manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
s, finite simplicial complex
Simplicial complex

In mathematics, a simplicial complex is a topological space of a particular kind, constructed by "gluing together" Point s, line segments, triangles, and their n-dimensional counterparts ....
es or CW complex
CW complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. The idea was to have a class of spaces that was broader than simplicial complexes , but still retained a combinatorial nature, so that computational considerations were not ignored....
es), the sequence of Betti numbers is 0 from some points onwards (Betti numbers vanish about the dimension of a space), and consists of natural numbers (i.e., finite, not infinity).

The term "Betti numbers" was coined by Henri Poincaré
Henri Poincaré

Jules Henri Poincar? was a French mathematician and theoretical physicist, and a philosophy of science. Poincar? is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime....
, the name being for Enrico Betti
Enrico Betti

Enrico Betti was an Italian mathematician, now remembered mostly for his 1871 paper on topology that led to the later naming after him of the Betti numbers....
.

Definition


For a non-negative integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
 , the -th Betti number of the space is defined as the rank
Rank of an abelian group

In mathematics, the rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space over the rational numbers one would need to "contain" it; or alternatively how large a free abelian group it can contain as a subgroup....
 of the abelian group
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
 , the -th homology group of . Equivalently, one can define it as the vector space dimension of , since the homology group in this case is a vector space over . The universal coefficient theorem
Universal coefficient theorem

In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A....
, in a very simple case, shows that these definitions are the same.

More generally, given a field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
  one can define , the -th Betti number with coefficients in , as the vector space dimension of .

Example: the first Betti number in graph theory


In topological graph theory
Topological graph theory

In mathematics topological graph theory is a branch of graph theory. It studies the embedding of graph s in surfaces, and graphs as topological spaces....
 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
Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then...
 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
Cyclomatic complexity

Cyclomatic complexity is a software metric . It was developed by Thomas J. McCabe in 1976 and is used to measure the complexity of a program. It directly measures the number of linearly independent paths through a program's source code....
 for an application of the first Betti number in software engineering
Software engineering

Software engineering is the application of a systematic, disciplined, quantifiable approach to the development, operation, and maintenance of software, and the study of these approaches....
.

Properties


The (rational) Betti numbers do not take into account any torsion
Torsion subgroup

In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order ....
 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
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
, 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 are finitely-generated, and so has a finite rank. Also the group is 0 when exceeds the top dimension of a simplex of .

For a finite CW-complex K we have where denotes Euler characteristic
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
 of K and any field F.

For any two spaces X and Y we have where denotes the Poincaré polynomial of X, (more generally, the Poincaré series, for infinite-dimensional spaces), i.e. the generating function
Generating function

In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers....
 of the Betti numbers of X: see Künneth theorem
Künneth theorem

In mathematics, especially in homological algebra and algebraic topology, a K?nneth theorem is a statement relating the homology of two objects to the homology of their product....
.

If X is n-dimensional manifold, there is symmetry interchanging k and n − k, for any k: under conditions (a closed and oriented manifold); see Poincaré duality
Poincaré duality

In mathematics, the Poincar? duality theorem, named after Henri Poincar?, is a basic result on the structure of the homology and cohomology group s of manifolds....
.

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
Universal coefficient theorem

In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A....
 (based on Tor functors, but in a simple case).

Examples


  1. The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...;
    the Poincaré polynomial is .
  2. The Betti number sequence for a two-torus
    Torus

    In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
     is 1, 2, 1, 0, 0, 0, ...;
    the Poincaré polynomial is .
  3. The Betti number sequence for a three-torus
    Torus

    In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
     is 1, 3, 3, 1, 0, 0, 0, ... .
    the Poincaré polynomial is .


Similarly, for an n-torus
Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
, the Poincaré polynomial is (by the Künneth theorem
Künneth theorem

In mathematics, especially in homological algebra and algebraic topology, a K?nneth theorem is a statement relating the homology of two objects to the homology of their product....
), so the Betti numbers are the binomial coefficient
Binomial coefficient

In mathematics, the binomial coefficient is the coefficient of the x k term in the polynomial expansion of the binomial exponentiation  n....
s.

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
Complex projective space

In mathematics, complex projective space, P, Pn or CPn, in fact preferablyis the projective space of line in Cn+1....
, with sequence 1, 0, 1, 0, 1, ... that is periodic, with period length 2. In this case the Poincaré function is not a polynomial but rather an infinite series , which, being a geometric series, can be expressed as the rational function .

More generally, any sequence that is periodic can be expressed as a sum of geometric series, generalizing the above (e.g., has generating function ), and more generally linear recursive sequences are exactly the sequences generated by rational functions; thus the Poincaré series is expressible as a rational function if and only if the sequence of Betti numbers is a linear recursive sequence.

Relationship with dimensions of spaces of differential form
Differential form

In the mathematics fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates....
s

In geometric situations when is a closed manifold
Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact space manifold without boundary. In contexts where no boundary is possible, any compact manifold 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
Modulo

The word modulo, in the mathematical community, is often used informally, in many imprecise ways. Generally, to say "A is the same as B modulo C" means, more-or-less, "A and B are the same except for differences accounted for or explained by C"....
 exact differential forms. The connection with the definition given above is via three basic results, de Rham's theorem and Poincaré duality
Poincaré duality

In mathematics, the Poincar? duality theorem, named after Henri Poincar?, is a basic result on the structure of the homology and cohomology group s of manifolds....
 (when those apply), and the universal coefficient theorem
Universal coefficient theorem

In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A....
 of homology theory
Homology theory

In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of Homology theories on topological spaces....
.

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
Hodge theory

In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a R...
, about the Hodge Laplacian.

In this setting, Morse theory
Morse theory

In differential topology, the techniques of Morse theory give a very direct way of analyzing the topological space of a manifold by studying differentiable functions on that manifold....
 gives a set of inequalities for alternating sums of Betti numbers in terms of a corresponding alternating sum of the number of critical points
Critical point (mathematics)

In mathematics, a critical point is a Point on the domain of a function of a function where:* one dimension: the derivative is equality to 0 or a point where the function ceases to be differentiable....
  of a Morse function of a given index
Morse theory

In differential topology, the techniques of Morse theory give a very direct way of analyzing the topological space of a manifold by studying differentiable functions on that manifold....
:

Witten
Edward Witten

Edward Witten is an United States theoretical physicist and professor at the Institute for Advanced Study. He is one of the world's leading researchers in superstring theory....
 gave an explanation of these inequalities by using the Morse function to modify the exterior derivative
Exterior derivative

In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a form of degree zero, to differential forms of higher degree....
 in the de Rham complex.