Betti number

Betti numbersPoincaré polynomialBettischen ZahlenCoining the term "Betti numberfirst Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.wikipedia
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Enrico Betti

BettiBetti, Enrico
The term "Betti numbers" was coined by Henri Poincaré after Enrico Betti.
Enrico Betti Glaoui (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.

Generating function

exponential generating functionordinary generating functiongenerating functions
The Poincaré polynomial of a surface is defined to be the generating function of its Betti numbers.
These are important in that many finite sequences can usefully be interpreted as generating functions, such as the Poincaré polynomial and others.

Algebraic topology

algebraicalgebraic topologistalgebraic topological
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.

Simplicial homology

simplicialsimplicial chain complexHomology
The n th Betti number represents the rank of the n th homology group, denoted H n, which tells us the maximum amount of cuts that must be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. These numbers are used today in fields such as simplicial homology, computer science, digital images, etc.
is called the kth Betti number of S.

Universal coefficient theorem

universal coefficient theorem for cohomology
The universal coefficient theorem, in a very simple torsion-free case, shows that these definitions are the same. 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.
Quite generally, the result indicates the relationship that holds between the Betti numbers

Component (graph theory)

connected componentconnected componentscomponent
In topological graph theory the first Betti number of a graph G with n vertices, m edges and k connected components equals
In topological graph theory it can be interpreted as the zeroth Betti number of the graph.

Poincaré duality

Poincaré dualPoincare dualitydual
under conditions (a closed and oriented manifold); see Poincaré duality. 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.
It was stated in terms of Betti numbers: The kth and (n-k)th Betti numbers of a closed (i.e., compact and without boundary) orientable n-manifold are equal.

Circuit rank

cyclomatic numbercyclotomic number
The first Betti number is also called the cyclomatic number—a term introduced by Gustav Kirchhoff before Betti's paper.
The circuit rank can be explained in terms of algebraic graph theory as the dimension of the cycle space of a graph, in terms of matroid theory as the corank of a graphic matroid, and in terms of topology as one of the Betti numbers of a topological space derived from the graph.

Cyclomatic complexity

Synchronization complexitycode complexitycomplex
See cyclomatic complexity for an application to software engineering.
In this case, the graph is strongly connected, and the cyclomatic complexity of the program is equal to the cyclomatic number of its graph (also known as the first Betti number), which is defined as :M = E − N + P.

Künneth theorem

Künneth formulaKünneth isomorphismKünneth
:see Künneth theorem.
A consequence of this result is that the Betti numbers, the dimensions of the homology with \Q coefficients, of X \times Y can be determined from those of X and Y.

Euler characteristic

Euler's formulaEuler–Poincaré characteristicElements
where \chi(K) denotes Euler characteristic of K and any field F.
More generally still, for any topological space, we can define the nth Betti number b n as the rank of the n-th singular homology group.

Henri Poincaré

PoincaréJules Henri PoincaréH. Poincaré
The term "Betti numbers" was coined by Henri Poincaré after Enrico Betti.

Complex projective space

complexcomplex projective 3-spaceC'''P 2
An example is the infinite-dimensional complex projective space, with sequence 1, 0, 1, 0, 1, ... that is periodic, with period length 2.
Therefore, the Betti numbers run

Homology (mathematics)

homologyhomology theoryhomology group
The n th Betti number represents the rank of the n th homology group, denoted H n, which tells us the maximum amount of cuts that must be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. These numbers are used today in fields such as simplicial homology, computer science, digital images, etc. 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. For a non-negative integer k, the kth Betti number b k (X) of the space X is defined as the rank (number of linearly independent generators) of the abelian group H k (X), the kth homology group of X.
The possible configurations of orientable cycles are classified by the Betti numbers of the manifold (Betti numbers are a refinement of the Euler characteristic).

Hodge theory

Hodge numberHodge decompositionharmonic form
There is an alternate reading, namely that the Betti numbers give the dimensions of spaces of harmonic forms.
The Betti numbers of X are the sum of the Hodge numbers in a given row.

Morse theory

Morse functionMorse lemmaMorse–Bott function
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:
Therefore, the rank of the γ th homology group,i.e., the Betti number b_\gamma(M), is less than or equal to the number of critical points of index γ of a Morse function on M.

De Rham cohomology

de Rham complexde Rham's theoremde Rham
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.
In particular, this implies that the 1st Betti number of a

Topological space

topologytopological spacestopological structure
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.

Simplicial complex

simplicial complexescomplexsimplicial
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.

Computer science

computer scientistcomputer sciencescomputer scientists
The n th Betti number represents the rank of the n th homology group, denoted H n, which tells us the maximum amount of cuts that must be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. These numbers are used today in fields such as simplicial homology, computer science, digital images, etc.

Digital image

digital imagesimagesimage
The n th Betti number represents the rank of the n th homology group, denoted H n, which tells us the maximum amount of cuts that must be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. These numbers are used today in fields such as simplicial homology, computer science, digital images, etc.

Emmy Noether

NoetherAmalie "Emmy" NoetherE. Noether
The modern formulation is due to Emmy Noether.

Zonal and meridional

meridionalzonalmeridional flow
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.

Integer

integersintegralZ
For a non-negative integer k, the kth Betti number b k (X) of the space X is defined as the rank (number of linearly independent generators) of the abelian group H k (X), the kth homology group of X.