Definition of the ring for a complex

Hitherto we have developed side by side the dual theories of homology and contrahomology, but have given little justification for introducing contrahomology. In chapter 7 we shall show, certainly, how the obstruction to extending a partial map of a polyhedron is naturally expressible as a contrahomology class; on the other hand, in chapter 5 we shall show that the integral homology groups of a finite complex determine the homology and contrahomology groups with arbitrary coefficients. The reader therefore may be disposed to think that everything could be done in terms of homology alone.

In this section, however, we enrich the concept of the contrahomology group by introducing a multiplication of contrahomology classes (provided the value group is a ring), which is topologically invariant and is preserved under the contrahomology homomorphisms induced by continuous maps. Under this multiplication the contrahomology classes form a ring and this ring cannot be deduced from the integral homology groups. The homology classes of a general complex appear not to have an invariant multiplicative structure; in the case of a complex whose polyhedron is an orientable manifold (locally Euclidean) the homology classes form the intersection ring, and a strict duality can then be established between the homology and contrahomology rings; in general, however, the contrahomology multiplication provides a structural element not available from the homology groups alone. We shall not be concerned with the intersection ring, preferring to deal with the more general contrahomology product.