Betti number

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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 points onward (Betti numbers vanish above the dimension of a space), and they are all finite.

A torus.
A torus has one connected component (b0), two circular holes (b1,the one in the center and the one in the middle of the "donut"), and one two-dimensional void (b2, the inside of the "donut") yielding Betti numbers of 1 (b0),2 (b1),1 (b2).

The n^{th} Betti number represents the rank of the n^{th} homology group, denoted Hn, which tells us the maximum amount of cuts that can 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.

Definition[edit]

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:

  • 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"

The two-dimensional Betti numbers are easily understandable because we see the world in 0, 1, 2, and 3-dimensions, however, the following Betti numbers are difficult to contemplate due to the higher-dimensional understanding.

For a non-negative integer k, the kth Betti number bk(X) of the space X is defined as the rank (number of generators) of the abelian group Hk(X), the kth homology group of X. The kth homology group is  H_{k} = ker \delta_{k}/im \delta_{k+1} , the  \delta_{k}s are the boundary maps of the simplicial complex and the rank of Hk is the kth Betti number. Equivalently, one can define it as the vector space dimension of Hk(XQ) 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 bk(XF), the kth Betti number with coefficients in F, as the vector space dimension of Hk(XF).

Example: Betti Numbers of a Simplicial Complex K[edit]

Let us go through a simple example of how to compute the Betti numbers for a simplicial complex. Example
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-simplice 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 shaded 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 \,

Properties[edit]

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.

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[edit]

  1. The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...;
    the Poincaré polynomial is
    1+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 \,.
  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)...(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^{2n-1})(1+x^3)(1+x^7)...(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[edit]

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.[2]

Uses in Mathematical Biology[edit]

In many biological settings, Betti numbers are used to understand the properties of genes located in breast cancer, by creating a curve of Betti numbers. For gene expression and copy number data sets, a graph is created out of one patients' log{2} ratios (y-axis), calculated in a DNA microarray, and the location of these log2 ratios in a specific chromosome (x-axis). Using a window of size 1,2,3,....,or n-dimensions, a point cloud can be constructed. For example, a size 2 window would mean that we take the log2 ratio of the first point from the start of the chromosome and that would be x1, while the second log2 ratio would be y1, repeating this process will produce y1 -> x1, y2 -> y1, ..., yn -> xn and y1 -> yn. Once the point cloud is created, a 1,2,3,....,or n-dimensional figure is made out of the data set and forms different simplicial complexes with a filtration (mathematics). The filtration is denoted by  \epsilon , which is a very small number, usually ranging from .0000001 to .1.  \epsilon is now considered the radius of a circle centered at each point in the point cloud. When two circles overlap, this forms a connection between the two points, as this process is continued, more simplices will show up with more Betti numbers as well. As  \epsilon increases, there are more "births" and "deaths" in the data, meaning that as the filtration changes, certain Betti numbers will decrease and others will increase. This method is used for copy number aberration(aCGH) and gene expression profiling data to indicate groups of patients, as opposed to looking at the individual patients. The significance in all of this comes from the hypothesis testing that is used to test the difference between subtypes of breast cancer. For example, testing whether the difference in b0 numbers between phenotypes HER2+/HER2- is 0 (HERneu). If the p-value calculated by this is close to 0, then the difference in the b0 curve for the two phenotypes is not close to 0. If the null hypothesis is not rejected, this would mean that the connected components would be the same or similar for the two different subtypes of breast cancer. Therefore, making the distance between the two types in this chromosome 0, concluding that no genes are significant or aberrant in this region. This same idea is used for the rest of the Betti numbers created by the filtration.

References[edit]

  1. ^ Barile, and Weisstein, Margherita and Eric. "Betti number". From MathWorld--A Wolfram Web Resource. 
  2. ^ 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 .