## Betti Numbers and Euler Characteristic

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 H_{q}
(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