Betti number
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 nth Betti number represents the rank of the nth homology group, denoted Hn, 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.
Betti numbers are used today in fields such as simplicial homology, computer science and digital images.
The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional simplicial complexes: Thus, for example, a torus has one connected surface component so b0 = 1, two "circular" holes (one equatorial and one meridional) so b1 = 2, and a single cavity enclosed within the surface so b2 = 1.
Another interpretation of bk 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 b1 = 2.
[2] The two-dimensional Betti numbers are easier to understand because we can see the world in 0, 1, 2, and 3-dimensions.
For a non-negative integer k, the kth Betti number bk(X) of the space X is defined as the rank (number of linearly independent generators) of the abelian group Hk(X), the kth homology group of X.
s are the boundary maps of the simplicial complex and the rank of Hk is the kth Betti number.
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 bk(X, F), the kth Betti number with coefficients in F, as the vector space dimension of Hk(X, F).
The Poincaré polynomial of a surface is defined to be the generating function of its Betti numbers.
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
Consider a 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: This may be proved straightforwardly by mathematical induction on the number of edges.
It is also called the cyclomatic number—a term introduced by Gustav Kirchhoff before Betti's paper.
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 (b0); one hole, which is the unshaded region (b1); and no "voids" or "cavities" (b2).
The Betti number sequence for this figure is 1, 1, 0, 0, ...; the Poincaré polynomial is
The finite component of the group is called the torsion coefficient of P. The (rational) Betti numbers bk(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.
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: see Künneth theorem.
: under conditions (a closed and oriented manifold); see Poincaré duality.
It is possible for spaces that are infinite-dimensional in an essential way to have an infinite sequence of non-zero Betti numbers.
The Poincaré polynomials of the compact simple Lie groups are: In geometric situations when
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.
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 of a given index: Edward Witten gave an explanation of these inequalities by using the Morse function to modify the exterior derivative in the de Rham complex.
