Betti number

These properties follow from the definition of B_k as the rank of the abelian group H_k(X), the k-th homology group of X. In the case of a simplicial complex this group is finitely-generated, and so has a finite rank. Also the group is 0 when k exceeds the top dimension of a simplex of X.

The Betti numbers do not take into account any torsion in the homology groups, but they are very useful basic topological invariants. For example the sequence for a circle is 1, 1, 0, 0, 0, ...; for a two-torus is 1, 2, 1, 0, 0, 0, ..., and for a three-torus is 1, 3, 3, 1, 0, 0, 0, ... . This is enough data to guess some important properties. For example, the behaviour for the Cartesian product XxY of spaces is expressed in this way: the generating function of the Betti numbers (called the Poincaré polynomial) multiplies. Therefore for an n-torus one should indeed see the binomial coefficients. Further there is symmetry interchanging k and n-k, for dimension n. This is a characteristic feature of the homology of a manifold, called Poincaré duality. As the names suggest, these ideas go back to Henri Poincaré.

Versions of these results are proved that include the torsion, too.

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 differential forms

For a differential manifold M, we can equip it with some auxiliary Riemannian metric. Then the Laplacian Δ, defined by *d*d using the exterior derivative and Hodge dual defines a homogeneous (in grading) linear differential operator acting upon the exterior algebra of differential forms: we can look at its action on each component of degree p separately.

If M is compact and oriented, the dimension of its kernel acting upon the space of p-forms is then equal (by Hodge theory) to that of the de Rham cohomology group in degree p: the Laplacian picks out a unique harmonic form in each cohomology class of closed forms.

The de Rham theorem tells us that the de Rham cohomology group is isomorphic with H^p(M;R). Then by Poincare duality for M, the dimension of this space is also the p^th Betti number. In summary, the Betti numbers in this case also count independent solutions of a Laplacian equation.

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