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The term "Betti numbers" was coined by Henri Poincaré and named for Enrico Betti .


DEFINITION


The ''k''-th Betti number

b


of the space ''X'' is defined as the Rank of the Abelian Group

H


the ''k''-th Homology Group of ''X''. Equivalently, one can define it as the Vector Space Dimension

of

H


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


the ''k''-th Betti number with coefficients in ''F'', as the vector space dimension
of

H



PROPERTIES


The 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 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 Hk(''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 imes Y}=P_XP_Y
where P_X denotes the Poincaré polynomial of ''X'', 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+...,
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 Betti number sequence for a two- Torus is 1, 2, 1, 0, 0, 0, ...;
#The Betti number sequence for a three- Torus is 1, 3, 3, 1, 0, 0, 0, ... .

In fact, for an ''n''- Torus one should indeed see the Binomial Coefficient s. This is a case of the Künneth theorem.

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.


RELATIONSHIP WITH DIMENSIONS OF SPACES OF DIFFERENTIAL FORM S


In geometric situations, 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 '' 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 also the use of some of the results of Hodge Theory , about the Hodge Laplacian .


BETTI NUMBER IN GRAPH THEORY


In Graph Theory the first Betti number of a graph ''G'' with ''n'' vertices,
''m'' edges and ''k'' connected components equals

m


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.