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.

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 formss.

The de Rham theorem tells us that the de Rham cohomology group is isomorphic with *H ^{p}*(