Editing Betti number

{{Short description|Roughly, the number of k-dimensional holes on a topological surface}}
In [[algebraic topology]], the '''Betti numbers''' are used to distinguish [[topological space]]s based on the connectivity of ''n''-dimensional [[simplicial complex]]es. For the most reasonable finite-dimensional [[topological space|space]]s (such as [[compact manifold]]s, 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 of a group|rank]] of the ''n''th [[homology group]], denoted ''H''<sub>''n''</sub>, 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.<ref>{{cite web|last=Barile, and Weisstein|first=Margherita and Eric|title=Betti number|url=http://mathworld.wolfram.com/BettiNumber.html|publisher=From MathWorld--A Wolfram Web Resource.}}</ref> For example, if <math>H_n(X) \cong 0</math> then <math>b_n(X) = 0</math>, if <math>H_n(X) \cong \mathbb{Z}</math> then <math>b_n(X) = 1</math>, if <math>H_n(X) \cong \mathbb{Z} \oplus \mathbb{Z}</math> then <math>b_n(X) = 2</math>, if <math>H_n(X) \cong \mathbb{Z} \oplus \mathbb{Z}\oplus \mathbb{Z}</math> then <math>b_n(X) = 3</math>, etc. Note that only the ranks of infinite groups are considered, so for example if  <math>H_n(X) \cong \mathbb{Z}^k \oplus \mathbb{Z}/(2)</math>, where <math>\mathbb{Z}/(2)</math> is the [[finite cyclic group]] of order 2, then <math>b_n(X) = k</math>. These finite components of the homology groups are their [[torsion subgroup]]s, and they are denoted by '''torsion coefficients'''.

The term "Betti number" was coined by [[Henri Poincaré]] after [[Enrico Betti]]. The modern formulation is due to [[Emmy Noether#Second epoch (1920–1926): Contributions to topology|Emmy Noether]]. Betti numbers are used today in fields such as [[simplicial homology]], [[computer science]] and [[digital images]].

== Geometric interpretation ==
[[File:Torus cycles.png|thumb|For a torus, the first Betti number is ''b''<sub>1</sub> = 2, which can be intuitively thought of as the number of circular "holes".]]
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]]es:
* ''b''<sub>0</sub> is the number of connected components;
* ''b''<sub>1</sub> is the number of one-dimensional or "circular" holes;
* ''b''<sub>2</sub> is the number of two-dimensional "voids" or "cavities".

Thus, for example, a torus has one connected surface component so ''b''<sub>0</sub> = 1, two "circular" holes (one equatorial and one [[Zonal and meridional|meridional]]) so ''b''<sub>1</sub> = 2, and a single cavity enclosed within the surface so ''b''<sub>2</sub> = 1.

Another interpretation of ''b''<sub>k</sub> 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''<sub>1</sub> = 2.<ref>Archived at [https://ghostarchive.org/varchive/youtube/20211212/XxFGokyYo6g Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20200829013025/https://www.youtube.com/watch?v=XxFGokyYo6g&gl=US&hl=en Wayback Machine]{{cbignore}}: {{Cite web|last=Albin|first=Pierre|date=2019|title=History of algebraic topology|website=[[YouTube]]|url=https://www.youtube.com/watch?v=XxFGokyYo6g}}{{cbignore}}</ref>

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]]&nbsp;''k'', the ''k''th Betti number ''b''<sub>''k''</sub>(''X'') of the space ''X'' is defined as the [[rank of an abelian group|rank]] (number of linearly independent generators) of the [[abelian group]] ''H''<sub>''k''</sub>(''X''), the ''k''th [[homology group]] of&nbsp;''X''. The ''k''th homology group is <math> H_{k} = \ker \delta_{k} / \operatorname{Im} \delta_{k+1} </math>,  the <math> \delta_{k}</math>s are the boundary maps of the [[simplicial complex]] and the rank of H<sub>k</sub> is the ''k''th Betti number. Equivalently, one can define it as the [[vector space dimension]] of ''H''<sub>''k''</sub>(''X'';&nbsp;'''Q''') since the homology group in this case is a vector space over&nbsp;'''Q'''. The [[universal coefficient theorem]], in a very simple torsion-free case, shows that these definitions are the same.

More generally, given a [[Field (mathematics)|field]] ''F'' one can define ''b''<sub>''k''</sub>(''X'',&nbsp;''F''), the ''k''th Betti number with coefficients in ''F'', as the vector space dimension of ''H''<sub>''k''</sub>(''X'',&nbsp;''F'').

== Poincaré polynomial ==
The '''Poincaré polynomial''' of a surface is defined to be the [[generating function]] of its Betti numbers.  For example, the Betti numbers of the torus are 1, 2, and 1; thus its Poincaré polynomial is <math>1+2x+x^2</math>. 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 <math>x^n</math> is <math>b_n</math>.

== Examples ==

=== Betti numbers of a graph ===
Consider a [[Topological graph theory|topological graph]] ''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:
: <math>H_k(G) = \begin{cases} 
  \mathbb Z^{|C|}         & k=0 \\
  \mathbb Z^{|E|+|C|-|V|} & k=1 \\
  \{0\}                   & \text{otherwise}
\end{cases}</math>

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.

Therefore, the "zero-th" Betti number ''b''<sub>0</sub>(''G'') equals |''C''|, which is simply the number of connected components.<ref name="Hage1996">{{cite book|author=Per Hage|url=https://books.google.com/books?id=ZBdLknuP0BYC&pg=PA49|title=Island Networks: Communication, Kinship, and Classification Structures in Oceania|publisher=Cambridge University Press|year=1996|isbn=978-0-521-55232-5|page=49}}</ref>

The first Betti number ''b''<sub>1</sub>(''G'') equals |''E''| + |''C''| - |''V''|. It is also called the [[cyclomatic number]]—a term introduced by [[Gustav Kirchhoff]] before Betti's paper.<ref name="Kotiuga2010">{{cite book|author=Peter Robert Kotiuga|url=https://books.google.com/books?id=mqLXi0FRIZwC&pg=PA20|title=A Celebration of the Mathematical Legacy of Raoul Bott|publisher=American Mathematical Soc.|year=2010|isbn=978-0-8218-8381-5|page=20}}</ref> See [[cyclomatic complexity]] for an application to [[software engineering]].

All other Betti numbers are 0.

=== Betti numbers of a simplicial complex ===
[[File:Simplicialexample.png|160x320px|alt=Example|right]]
Consider a [[simplicial complex]] 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. There is one connected component in this figure (''b''<sub>0</sub>); one hole, which is the unshaded region (''b''<sub>1</sub>); and no "voids" or "cavities" (''b''<sub>2</sub>).

This means that the rank of <math>H_0</math> is 1, the rank of <math>H_{1}</math> is 1 and the rank of <math>H_2</math> is 0.

The Betti number sequence for this figure is 1, 1, 0, 0, ...; the Poincaré polynomial is <math>1 + x\,</math>.

=== Betti numbers of the projective plane ===
The homology groups of the [[projective plane]] ''P'' are:
: <math>H_k(P) = \begin{cases} \mathbb Z & k=0 \\ \mathbb Z _ 2 & k=1 \\ \{0\} & \text{otherwise} \end{cases}</math>

Here, '''Z'''<sub>2</sub> is the [[cyclic group]] of order 2. The 0-th Betti number is again 1. However, the 1-st Betti number is 0. This is because ''H''<sub>1</sub>(''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''<sub>''k''</sub>(''X'') do not take into account any [[torsion subgroup|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
: <math>\chi(K) = \sum_{i=0}^\infty(-1)^i b_i(K, F), \,</math>
where <math>\chi(K)</math> denotes [[Euler characteristic]] of ''K'' and any field&nbsp;''F''.

=== Cartesian product ===
For any two spaces ''X'' and ''Y'' we have
: <math>P_{X\times Y} = P_X P_Y ,</math>
where <math>P_X</math> denotes the '''Poincaré polynomial''' of ''X'', (more generally, the [[Hilbert–Poincaré series]], for infinite-dimensional spaces), i.e., the [[generating function]] of the Betti numbers of ''X'':
: <math>P_X(z) = b_0(X) + b_1(X)z + b_2(X)z^2 + \cdots , \,\!</math>
see  [[Künneth theorem]].

=== Symmetry ===
If ''X'' is ''n''-dimensional manifold, there is symmetry interchanging <math>k</math> and <math>n - k</math>, for any <math>k</math>:
: <math>b_k(X) = b_{n-k}(X),</math>
under conditions (a ''closed'' and ''oriented'' manifold); see [[Poincaré duality]].

=== Different coefficients ===
The dependence on the field ''F'' is only through its [[characteristic (field)|characteristic]]. If the homology groups are [[torsion (algebra)|torsion-free]], the Betti numbers are independent of ''F''. The connection of ''p''-torsion and the Betti number for [[characteristic p|characteristic&nbsp;''p'']], for ''p'' a prime number, is given in detail by the [[universal coefficient theorem]] (based on [[Tor functor]]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
#:: <math>1 + x\,</math>.
# The Betti number sequence for a three-[[torus]] is 1, 3, 3, 1, 0, 0, 0, ... .
#: the Poincaré polynomial is
#:: <math>(1 + x)^3 = 1 + 3x + 3x^2 + x^3\,</math>.
# Similarly, for an ''n''-[[torus]],
#: the Poincaré polynomial is
#:: <math>(1 + x)^n \,</math> (by the [[Künneth theorem]]), so the Betti numbers are the [[binomial coefficient]]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]], 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 
: <math>1 + x^2 + x^4 + \dotsb</math>,
which, being a geometric series, can be expressed as the rational function
: <math>\frac{1}{1 - x^2}.</math>

More generally, any sequence that is periodic can be expressed as a sum of geometric series, generalizing the above. For example <math>a,b,c,a,b,c,\dots,</math> has the generating function 
: <math>\left(a + bx + cx^2\right)/\left(1 - x^3\right) \,</math>
and more generally [[linear recursive sequence]]s 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]] are:
: <math>\begin{align}
   P_{SU(n+1)}(x) &= \left(1 + x^3\right)\left(1 + x^5\right)\cdots\left(1 + x^{2n+1}\right) \\
  P_{SO(2n+1)}(x) &= \left(1 + x^3\right)\left(1 + x^7\right)\cdots\left(1 + x^{4n-1}\right) \\
     P_{Sp(n)}(x) &= \left(1 + x^3\right)\left(1 + x^7\right)\cdots\left(1 + x^{4n-1}\right) \\
    P_{SO(2n)}(x) &= \left(1 + x^{2n-1}\right)\left(1 + x^3\right)\left(1 + x^7\right)\cdots\left(1 + x^{4n-5}\right) \\
       P_{G_2}(x) &= \left(1 + x^3\right)\left(1 + x^{11}\right) \\
       P_{F_4}(x) &= \left(1 + x^3\right)\left(1 + x^{11}\right)\left(1 + x^{15}\right)\left(1 + x^{23}\right) \\
       P_{E_6}(x) &= \left(1 + x^3\right)\left(1 + x^{9}\right)\left(1 + x^{11}\right)\left(1 + x^{15}\right)\left(1 + x^{17}\right)\left(1 + x^{23}\right) \\
       P_{E_7}(x) &= \left(1 + x^3\right)\left(1 + x^{11}\right)\left(1 + x^{15}\right)\left(1 + x^{19}\right)\left(1 + x^{23}\right)\left(1 + x^{27}\right)\left(1 + x^{35}\right) \\
     P_{E_{8}}(x) &= \left(1 + x^3\right)\left(1 + x^{15}\right)\left(1 + x^{23}\right)\left(1 + x^{27}\right)\left(1 + x^{35}\right)\left(1 + x^{39}\right)\left(1 + x^{47}\right)\left(1 + x^{59}\right)
\end{align}</math>

== Relationship with dimensions of spaces of differential forms ==
In geometric situations when <math>X</math> 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 form]]s ''[[Modular arithmetic|modulo]]'' [[exact differential form]]s. 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 form]]s. This requires the use of some of the results of [[Hodge theory]] 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 point (mathematics)|critical points]] <math>N_i</math> of a [[Morse function]] of a given [[Morse theory|index]]:
: <math> b_i(X) - b_{i-1} (X) +  \cdots \le N _i - N_{i-1} + \cdots. </math>

[[Edward Witten]] gave an explanation of these inequalities by using the Morse function to modify the [[exterior derivative]] in the [[de Rham complex]].<ref>{{citation |last=Witten |first= Edward |author-link=Edward Witten |year=1982 |title=Supersymmetry and Morse theory |journal=[[Journal of Differential Geometry]] |volume=17 |issue=4 |pages=661–692 |doi=10.4310/jdg/1214437492 |doi-access=free }}{{open access}}</ref>

== See also ==
* [[Topological data analysis]]
* [[Torsion coefficient (topology)|Torsion coefficient]]
* [[Euler characteristic]]

== References ==
{{reflist}}

* {{citation |first=Frank Wilson |last=Warner |title=Foundations of differentiable manifolds and Lie groups |location=New York |publisher=Springer |year=1983 |isbn=0-387-90894-3 }}.
* {{citation |first=John |last=Roe |title=Elliptic Operators, Topology, and Asymptotic Methods |edition=Second |series=Research Notes in Mathematics Series |volume=395 |location=Boca Raton, FL |publisher=Chapman and Hall |year=1998 |isbn=0-582-32502-1 }}.

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