the Eilenberg-Steenrod axioms
/ EYE-len-berg STEEN-rod /
By the 1940s there were several constructions all called 'homology' — singular, simplicial, Cech — and they kept giving the same answers on reasonable spaces. Eilenberg and Steenrod asked the structural question: what minimal list of properties forces this agreement? Their axioms are the answer. They define what it even means to be a homology theory, and they show that on the category of CW pairs the axioms pin the theory down uniquely.
A homology theory is a sequence of functors H_n(X, A) (with H_n(X) the case A empty) together with connecting homomorphisms, required to satisfy: (1) Homotopy — homotopic maps induce the same map on homology. (2) Excision — the excision isomorphism holds. (3) The long exact sequence of a pair — the sequence ... -> H_n(A) -> H_n(X) -> H_n(X,A) -> H_{n-1}(A) -> ... is exact and natural. (4) Additivity — homology of a disjoint union is the direct sum of the pieces' homologies. And the pivotal (5) Dimension axiom — the homology of a single point is concentrated in degree zero, H_0(point) = G (the coefficient group) and H_n(point) = 0 for n nonzero. The theorem of Eilenberg-Steenrod: on CW pairs, any two homology theories with the same coefficient group and satisfying all five are naturally isomorphic.
The conceptual gain is enormous. You no longer have to prove singular and cellular and simplicial homology agree by hand each time; you check each satisfies the axioms and uniqueness does the rest. The axioms also reorganize the subject: every theorem about homology either follows from the axioms (so it holds for every homology theory) or genuinely uses the specific construction.
The deep twist is that the dimension axiom is the one you can drop. A theory satisfying axioms 1 through 4 but with H_n(point) allowed to be nonzero for n nonzero is called a generalized (or extraordinary) homology theory — topological K-theory, cobordism, and stable homotopy are exactly these. So the Eilenberg-Steenrod framework is not just a characterization of ordinary homology; by deleting one axiom it opens the door to the richest invariants in modern topology. Calling something 'a homology theory' without specifying the dimension axiom is genuinely ambiguous.
Using only the axioms, compute H_n(S^k). The dimension axiom gives the homology of a point; the long exact sequence and excision relate H_n(S^k) to H_{n-1}(S^{k-1}) via the suspension isomorphism; induction from S^0 (two points) yields H_n(S^k) = Z for n = 0, k and zero otherwise — derived without ever touching a singular simplex.
The sphere's homology follows from the axioms alone, with no specific model.
Dropping only the dimension axiom yields generalized homology theories such as K-theory and cobordism — these have nontrivial homology of a point. So 'homology theory' is ambiguous unless you say whether the dimension axiom is assumed.