Betti Number - Properties

Properties

The (rational) 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 to 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 H_k(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

where denotes Euler characteristic of K and any field F.

For any two spaces X and Y we have

where P_X 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:

see Künneth theorem.

If X is n-dimensional manifold, there is symmetry interchanging k and n − k, for any k:

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).