Betti number

In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (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 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.^[1] These numbers are used today in fields such as simplicial homology, computer science, digital images, etc.

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

Definitions

Informally, the kth Betti number refers to the number of k-dimensional holes on a topological surface. The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional simplicial complexes:

A torus has Betti numbers b₀ = 1, b₁ = 2, and b₂ = 1

b₀ is the number of connected components
b₁ is the number of one-dimensional or "circular" holes
b₂ is the number of two-dimensional "voids" or "cavities"

Thus, for example, a torus has one connected surface component so b₀ = 1, a circular (two-dimensional) hole at each end of the central space so b₁ = 2 and a single cavity enclosed within the surface so b₂ = 1.

Closely related to the Betti numbers of a topological surface is the Poincaré polynomial of that surface. 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 $1+2x+x^{2}$ . The same definition applies to any topological space which has a finitely generated homology.

The two-dimensional Betti numbers are easier to understand because we see the world in 0, 1, 2, and 3-dimensions, however. The following Betti numbers are higher-dimensional than apparent physical space.

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 kth homology group is $H_{{k}}=\ker \delta _{{k}}/{\mathrm {Im}}\delta _{{k+1}}$ , the $\delta _{{k}}s$ are the boundary maps of the simplicial complex and the rank of H_k is the kth 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 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 kth Betti number with coefficients in F, as the vector space dimension of H_k(X, F).

Given a topological space which has finitely generated homology, the Poincaré polynomial is defined as the generating function of its Betti numbers, viz the polynomial where the coefficient of $x^{n}$ is $b_{n}$ .

Example 1: Betti numbers of a simplicial complex K

Let us go through a simple example of how to compute the Betti numbers for a simplicial complex.

Here we have 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. It is clear that there is one connected component in this figure (b₀); one hole, which is the unshaded region (b₁); and no "voids" or "cavities" (b₂).

This means that the rank of $H_{0}$ is 1, the rank of $H_{{1}}$ is 1 and the rank of $H_{2}$ is 0.

The Betti number sequence for this figure is 1,1,0,0,...; the Poincaré polynomial is $1+x\,$

Example 2: the first 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

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.

The first Betti number is also called the cyclomatic number—a term introduced by Gustav Kirchhoff before Betti's paper.^[2] See cyclomatic complexity for an application to software engineering.

The "zero-th" Betti number of a graph is simply the number of connected components k.^[3]

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 to count the number of holes of different dimensions.

For a finite CW-complex K we have

\chi (K)=\sum _{{i=0}}^{\infty }(-1)^{i}b_{i}(K,F),\,

where $\chi (K)$ denotes Euler characteristic of K and any field F.

For any two spaces X and Y we have

P_{{X\times Y}}=P_{X}P_{Y},\,

where P_X denotes the Poincaré polynomial of X, (more generally, the Poincaré series, for infinite-dimensional spaces), i.e. the generating function of the Betti numbers of X:

P_{X}(z)=b_{0}(X)+b_{1}(X)z+b_{2}(X)z^{2}+\cdots ,\,\!

see Künneth theorem.

If X is n-dimensional manifold, there is symmetry interchanging k and n − k, for any k:

b_{k}(X)=b_{{n-k}}(X),\,\!

under conditions (a closed and oriented manifold); see Poincaré duality.

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).

Examples

The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...;
the Poincaré polynomial is
$1+x\,$ .
The Betti number sequence for a three-torus is 1, 3, 3, 1, 0, 0, 0, ... .
the Poincaré polynomial is
$(1+x)^{3}=1+3x+3x^{2}+x^{3}\,$ .
Similarly, for an n-torus,
the Poincaré polynomial is
$(1+x)^{n}\,$ (by the Künneth theorem), so the Betti numbers are the binomial coefficients.

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

1+x^{2}+x^{4}+\dotsb

which, being a geometric series, can be expressed as the rational function

{\frac {1}{1-x^{2}}}.

More generally, any sequence that is periodic can be expressed as a sum of geometric series, generalizing the above (e.g., $a,b,c,a,b,c,\dots ,$ has generating function

(a+bx+cx^{2})/(1-x^{3})\,

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 are:

P_{{SU(n+1)}}(x)=(1+x^{3})(1+x^{5})\cdots (1+x^{{2n+1}})

P_{{SO(2n+1)}}(x)=(1+x^{3})(1+x^{7})\cdots (1+x^{{4n-1}})

P_{{Sp(n)}}(x)=(1+x^{3})(1+x^{7})\cdots (1+x^{{4n-1}})

P_{{SO(2n)}}(x)=(1+x^{{2n-1}})(1+x^{3})(1+x^{7})\cdots (1+x^{{4n-5}})

P_{{G_{2}}}(x)=(1+x^{3})(1+x^{{11}})

P_{{F_{4}}}(x)=(1+x^{3})(1+x^{{11}})(1+x^{{15}})(1+x^{{23}})

P_{{E_{6}}}(x)=(1+x^{3})(1+x^{{9}})(1+x^{{11}})(1+x^{{15}})(1+x^{{17}})(1+x^{{23}})

P_{{E_{7}}}(x)=(1+x^{3})(1+x^{{11}})(1+x^{{15}})(1+x^{{19}})(1+x^{{23}})(1+x^{{27}})(1+x^{{35}})

P_{{E_{{8}}}}(x)=(1+x^{3})(1+x^{{15}})(1+x^{{23}})(1+x^{{27}})(1+x^{{35}})(1+x^{{39}})(1+x^{{47}})(1+x^{{59}})

Relationship with dimensions of spaces of differential forms

In geometric situations when $X$ 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 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.

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 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:

b_{i}(X)-b_{{i-1}}(X)+\cdots \leq N_{i}-N_{{i-1}}+\cdots .

Witten gave an explanation of these inequalities by using the Morse function to modify the exterior derivative in the de Rham complex.^[4]

References

↑ Barile, and Weisstein, Margherita and Eric. "Betti number". From MathWorld--A Wolfram Web Resource.
↑ Peter Robert Kotiuga (2010). A Celebration of the Mathematical Legacy of Raoul Bott. American Mathematical Soc. p. 20. ISBN 978-0-8218-8381-5.
↑ Per Hage (1996). Island Networks: Communication, Kinship, and Classification Structures in Oceania. Cambridge University Press. p. 49. ISBN 978-0-521-55232-5.
↑ Witten, Edward (1982). Supersymmetry and Morse theory. J. Differential Geom. 17 (1982), no. 4, 661–692.

Warner, Frank Wilson (1983), Foundations of differentiable manifolds and Lie groups, New York: Springer, ISBN 0-387-90894-3 .
Roe, John (1998), Elliptic Operators, Topology, and Asymptotic Methods, Research Notes in Mathematics Series, 395 (Second ed.), Boca Raton, FL: Chapman and Hall, ISBN 0-582-32502-1 .

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