The Eilenberg–Steenrod axioms apply to a sequence of functors from the category of pairs of topological spaces to the category of abelian groups, together with a natural transformation called the boundary map (here is a shorthand for ). The axioms are:
- Homotopy: Homotopic maps induce the same map in homology. That is, if is homotopic to , then their induced homomorphisms are the same.
- Excision: If is a pair and U is a subset of A such that the closure of U is contained in the interior of A, then the inclusion map induces an isomorphism in homology.
- Dimension: Let P be the one-point space; then for all .
- Additivity: If , the disjoint union of a family of topological spaces , then
- Exactness: Each pair (X, A) induces a long exact sequence in homology, via the inclusions and :
If P is the one point space, then is called the coefficient group. For example, singular homology (taken with integer coefficients, as is most common) has as coefficients the integers.
Some facts about homology groups can be derived directly from the axioms, such as the fact that homotopically equivalent spaces have isomorphic homology groups.
The homology of some relatively simple spaces, such as n-spheres, can be calculated directly from the axioms. From this it can be easily shown that the (n − 1)-sphere is not a retract of the n-disk. This is used in a proof of the Brouwer fixed point theorem.