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. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

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. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... The infinity symbol ∞ in several typefaces. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ... 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 term "Betti numbers" was coined by Henri Poincaré, the name being for Enrico Betti. Jules Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [][1]), was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Enrico Betti (21 October 1823 - 11 August 1892) 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

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 , since the homology group in this case is a vector space over . The universal coefficient theorem, in a very simple case, shows that these definitions are the same. 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 one would need to contain it; or alternatively how large a free abelian group it can contain as a subgroup. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, the dimension of a vector space V is the cardinality (i. ... 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. It shows that the integral homology groups Hi(X,Z) do in a certain, definite sense...

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). In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

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 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 group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... Circle illustration In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... A modern factory produced hard pretzel. ...

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 k exceeds the top dimension of a simplex of X.

For a finite CW-complex K we have

where χ(K) denotes Euler characteristic of K and any field F. It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...

For any two spaces X and Y we have

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

see Künneth theorem. In mathematics, the Künneth theorem of algebraic topology describes the singular homology of the cartesian product X × Y of two topological spaces, in terms of singular homology groups Hi(X, R) and Hj(X, R). ...

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. In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups 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 (based on Tor functors, but in a simple case). In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0 (where n1R is defined as 1R + ... + 1R with n summands). ... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... 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. It shows that the integral homology groups Hi(X,Z) do in a certain, definite sense... The Tor functors are the derived functors of the tensor product functor in mathematics. ...

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. This is a case of the Künneth theorem. A torus. ... A torus. ... A torus. ... In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist. ...

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. In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. ... In mathematics, a periodic function is a function that repeats its values, after adding some definite period to the variable. ...

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. A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given... The word modulo (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means a small measure. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given... In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ... In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. ... 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. It shows that the integral homology groups Hi(X,Z) do in a certain, definite sense... In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles 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, about the Hodge Laplacian. In mathematics, Hodge theory is the study of the consequences for the algebraic topology of a smooth manifold M of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M. It was developed by W. V. D. Hodge in the 1930s as an extension... In mathematics, Hodge theory 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 Riemannian metric... In mathematics and physics, the Laplace operator or Laplacian, denoted by   or   and named after Pierre-Simon Laplace, is a differential operator, specifically an important case of an elliptic operator, with many applications. ...

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 In mathematics topological graph theory is a branch of graph theory. ...

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. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...