Betti number

In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces.

Each Betti number is a natural number or +∞. 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 points onwards (Betti numbers vanish above the dimension of a space), and they are all finite.

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

Informal Definition
A torus.
A torus has one connected component, two circular holes (the one in the center and the one in the middle of the "tube"), and one three-dimensional void (the inside of the "tube") yielding Betti numbers of 1,2,1.

Informally, the kth Betti number refers to the number of unconnected k-dimensional surfaces.[1] The first few Betti numbers have the following intuitive definitions:

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

Definition

For a non-negative integer k, the kth Betti number b_k(X) of the space X is defined as the rank of the abelian group H_k(X), the kth homology group of X. 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 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).

Example: 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.

See cyclomatic complexity for an application of the first Betti number in software engineering.

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 circle, the first Betti number is 1. For a general pretzel, the first Betti number is twice the number of holes.

In the case of a finite simplicial complex the homology groups H_k(X, Z) 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

\( \chi(K)=\sum_{i=0}^\infty(-1)^ib_i(K,F), \, \)

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 PX 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 two-torus is 1, 2, 1, 0, 0, 0, ...;

the Poincaré polynomial is

\( (1+x)^2=1+2x+x^2 \,. \)

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

\( \frac{1}{1-x^2}=1+x^2+(x^2)^2+(x^2)^3+\dotsb. \)

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:

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

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

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

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

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

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

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

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

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

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:

\( b_i(X) - b_{i-1} (X) + \cdots \le N _i - N_{i-1} + \cdots. \)

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

^ Carlsson, G. “Topology and data.” AMERICAN MATHEMATICAL SOCIETY 46.2 (2009): 255-308.

Warner, Frank Wilson (1983), Foundations of differentiable manifolds and Lie groups, New York: Springer, ISBN 0387908943.
Roe, John (1998), Elliptic Operators, Topology, and Asymptotic Methods, Research Notes in Mathematics Series, 395 (Second ed.), Boca Raton, FL: Chapman and Hall, ISBN 0582325021.

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