In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common.
The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod.
One can define a homology theory as a sequence of functors satisfying the Eilenberg–Steenrod axioms.
The axiomatic approach, which was developed in 1945, allows one to prove results, such as the Mayer–Vietoris sequence, that are common to all homology theories satisfying the axioms.
Extraordinary cohomology theories first arose in K-theory and cobordism.
of topological spaces to the category of abelian groups, together with a natural transformation
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.
This is used in a proof of the Brouwer fixed point theorem.