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In algebraic topology, the Betti numbers of a
topological space X are a sequence B0,
B1, ... of topological invariants. Each Betti number is a natural number, or infinity. For the most reasonable spaces
(such as compact manifolds, finite
simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some points onwards, and consists of natural numbers. The
name is for Enrico Betti.
These properties follow from the definition of Bk(X) as the rank of the abelian group
Hk(X), the k-th homology group of X.
Equivalently one can define it as dimension of Hk(X,Q) since the homology group in this case
is a vector space over Q. More generally, given a field F one can define Bk(X,F) (the k-th Betti number
with respect to F) as the dimension of Hk(X,F).
In the case of a finite 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 a finite CW-complex K we have
-
where χ(K) denotes Euler characteristic of K and any field F.
Examples
The Betti number 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 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.
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.
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