III.38 Homology and Cohomology
If X is a TOPOLOGICAL SPACE [III.90], then one can associate with it a sequence of groups Hn (X, R), where R is a commutative RING [III.81 §1] such as 
or 
. These groups, the homology groups of X (with coefficients in R), are a powerful invariant: powerful because they contain a great deal of information about X but are nevertheless easy to compute, at least compared with some other invariants. The closely related cohomology groups Hn(X, R) are more useful still because they can be made into a ring: to oversimplify slightly, an element of the cohomology group Hn(X) is an EQUIVALENCE CLASS [I.2 §2.3] [Y] of a subspace Y of codimension n. (Of course, for this to make true sense X should be a fairly nice space such as a MANIFOLD [I.3 §6.9].) Then, if [Y] and [Z] belong to Hn(X, R) and Hm(X, R), respectively, their product is [Y ∩ Z]. Since Y ∩ Z “typically” has codimension n + m, the equivalence class [Y ∩ Z] belongs to Hn+m (X, R). Homology and cohomology groups are described in more detail in ALGEBRAIC TOPOLOGY [IV.6].
The concepts of homology and cohomology have become far more general than the above discussion suggests, and are no longer tied to topological spaces: for instance, the notion of group cohomology is of great importance in algebra. Even within topology, there are many different homology and cohomology theories. In 1945, Eilenberg and Steenrod devised a small number of axioms that greatly clarified the area: a homology theory is any association of groups with topological spaces that satisfies these axioms, and the fundamental properties of homology theories follow from the axioms.