In
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, the Betti numbers are used to distinguish
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s based on the connectivity of ''n''-dimensional
simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
es. For the most reasonable finite-dimensional
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
s (such as
compact manifolds, finite
simplicial complexes or
CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite.
The ''n''
th Betti number represents the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
of the ''n''
th homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
, denoted ''H''
''n'', which tells us the maximum number of cuts that can be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. For example, if
then
, if
then
, if
then
, if
then
, etc. Note that only the ranks of infinite groups are considered, so for example if
, where
is the finite cyclic group of order 2, then
. These finite components of the homology groups are their
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 (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
s, and they are denoted by torsion coefficients.
The term "Betti numbers" was coined by
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...
after
Enrico Betti. The modern formulation is due to
Emmy Noether
Amalie Emmy NoetherEmmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...
. Betti numbers are used today in fields such as
simplicial homology,
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
,
digital images
A digital image is an image composed of picture elements, also known as ''pixels'', each with '' finite'', '' discrete quantities'' of numeric representation for its intensity or gray level that is an output from its two-dimensional functions ...
, etc.
Geometric interpretation
Informally, the ''k''th Betti number refers to the number of ''k''-dimensional ''holes'' on a topological surface. A "''k''-dimensional ''hole''" is a ''k''-dimensional cycle that is not a boundary of a (''k''+1)-dimensional object.
The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional
simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
es:
* ''b''
0 is the number of connected components;
* ''b''
1 is the number of one-dimensional or "circular" holes;
* ''b''
2 is the number of two-dimensional "voids" or "cavities".
Thus, for example, a torus has one connected surface component so ''b''
0 = 1, two "circular" holes (one equatorial and one
meridional) so ''b''
1 = 2, and a single cavity enclosed within the surface so ''b''
2 = 1.
Another interpretation of ''b''
k is the maximum number of ''k''-dimensional curves that can be removed while the object remains connected. For example, the torus remains connected after removing two 1-dimensional curves (equatorial and meridional) so ''b''
1 = 2.
The two-dimensional Betti numbers are easier to understand because we can see the world in 0, 1, 2, and 3-dimensions.
Formal definition
For a non-negative
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
''k'', the ''k''th Betti number ''b''
''k''(''X'') of the space ''X'' is defined as the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
(number of linearly independent generators) of the
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
''H''
''k''(''X''), the ''k''th
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
of ''X''. The ''k''th homology group is
, the
s are the boundary maps of the
simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
and the rank of H
k is the ''k''th Betti number. Equivalently, one can define it as the
vector space dimension of ''H''
''k''(''X''; Q) since the homology group in this case is a vector space over Q. The
universal coefficient theorem
In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'':
:
completely ...
, in a very simple torsion-free 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'').
Poincaré polynomial
The Poincaré polynomial of a surface is defined to be the
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
of its Betti numbers. For example, the Betti numbers of the torus are 1, 2, and 1; thus its Poincaré polynomial is
. The same definition applies to any topological space which has a finitely generated homology.
Given a topological space which has a finitely generated homology, the Poincaré polynomial is defined as the generating function of its Betti numbers, via the polynomial where the coefficient of
is
.
Examples
Betti numbers of a graph
Consider a
topological graph
In mathematics, a topological graph is a representation of a graph in the plane, where the ''vertices'' of the graph are represented by distinct points and the ''edges'' by Jordan arcs (connected pieces of ''Jordan curves'') joining the corre ...
''G'' in which the set of vertices is ''V'', the set of edges is ''E'', and the set of connected components is ''C''. As explained in the page on
graph homology, its homology groups are given by:
:
This may be proved straightforwardly by
mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
on the number of edges. A new edge either increments the number of 1-cycles or decrements the number of connected components.
Therefore, the "zero-th" Betti number ''b''
0(''G'') equals , ''C'', , which is simply the number of connected components.
The first Betti number ''b''
1(''G'') equals , ''E'', + , ''C'', - , ''V'', . It is also called the
cyclomatic number—a term introduced by
Gustav Kirchhoff
Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects.
He ...
before Betti's paper.
See
cyclomatic complexity
Cyclomatic complexity is a software metric used to indicate the complexity of a program. It is a quantitative measure of the number of linearly independent paths through a program's source code. It was developed by Thomas J. McCabe, Sr. in 1976. ...
for an application to
software engineering
Software engineering is a systematic engineering approach to software development.
A software engineer is a person who applies the principles of software engineering to design, develop, maintain, test, and evaluate computer software. The term '' ...
.
All other Betti numbers are 0.
Betti numbers of a simplicial complex
Consider a
simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
with 0-simplices: a, b, c, and d, 1-simplices: E, F, G, H and I, and the only 2-simplex is J, which is the shaded region in the figure. It is clear that there is one connected component in this figure (''b''
0); one hole, which is the unshaded region (''b''
1); and no "voids" or "cavities" (''b''
2).
This means that the rank of
is 1, the rank of
is 1 and the rank of
is 0.
The Betti number sequence for this figure is 1, 1, 0, 0, ...; the Poincaré polynomial is
.
Betti numbers of the projective plane
The homology groups of the
projective plane ''P'' are:
:
Here, Z
2 is the
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order 2. The 0-th Betti number is again 1. However, the 1-st Betti number is 0. This is because ''H''
1(''P'') is a finite group - it does not have any infinite component. The finite component of the group is called the torsion coefficient of ''P''. 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.
Properties
Euler characteristic
For a finite CW-complex ''K'' we have
:
where
denotes
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
of ''K'' and any field ''F''.
Cartesian product
For any two spaces ''X'' and ''Y'' we have
:
where
denotes the Poincaré polynomial of ''X'', (more generally, the
Hilbert–Poincaré series, for infinite-dimensional spaces), i.e., the
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
of the Betti numbers of ''X'':
:
see
Künneth theorem
In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their product. The classical statement of the K� ...
.
Symmetry
If ''X'' is ''n''-dimensional manifold, there is symmetry interchanging
and
, for any
:
:
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. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
.
Different coefficients
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
In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'':
:
completely ...
(based on
Tor functor
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to co ...
s, but in a simple case).
More examples
# The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...;
#: the Poincaré polynomial is
#::
.
# The Betti number sequence for a three-
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
is 1, 3, 3, 1, 0, 0, 0, ... .
#: the Poincaré polynomial is
#::
.
# Similarly, for an ''n''-
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
,
#: the Poincaré polynomial is
#::
(by the
Künneth theorem
In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their product. The classical statement of the K� ...
), so the Betti numbers are the
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
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
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
, 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. For example
has the 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.
The Poincaré polynomials of the compact simple
Lie groups
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
are:
:
Relationship with dimensions of spaces of differential forms
In geometric situations when
is a
closed manifold
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example i ...
, 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
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
''
exact differential forms. The connection with the definition given above is via three basic results,
de Rham's theorem and
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. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
(when those apply), and the
universal coefficient theorem
In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'':
:
completely ...
of
homology theory
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
.
There is an alternate reading, namely that the Betti numbers give the dimensions of spaces of
harmonic form
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coh ...
s. This requires the use of some of the results of
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every co ...
on 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 of a
Morse function
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
of a given
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
:
:
Edward Witten
Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, q ...
gave an explanation of these inequalities by using the Morse function to modify the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
in the
de Rham complex.
See also
*
Topological data analysis
*
Torsion coefficient
*
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
References
*.
*.
{{DEFAULTSORT:Betti Number
Algebraic topology
Graph invariants
Topological graph theory
Generating functions