ABSTRACT
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 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429502408/295d3acc-dfa3-4421-a2d3-a9c22ffb681c/content/eq578.tif"/>
