chapter  20
6 Pages

Betti Numbers and Euler Characteristic

WithMarvin J. Greenberg, John R. Harper

If we take homology with integer coefficients, for certain spaces, such as spherical complexes (19.20), the homology groups will be finitely generated. If A is a finitely generated Abelian group, a basic theorem (see Lang [35], p. 45) states that the elements of finite order in A form the torsion subgroup T and that the quotient group A/T is free Abelian. The minimal number of generators of A/T is called the rank of A. The rank of Hq (X; Z) is called the q-th Betti number β;q of the space X, and we also define the Euler characteristic χ (X) by the formula χ ( X ) = ∑ q ( − 1 ) q β q