Eilenberg–Steenrod axioms

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.

If one omits the dimension axiom (described below), then the remaining axioms define what is called an extraordinary homology theory. Extraordinary cohomology theories first arose in K-theory and cobordism.

The Eilenberg–Steenrod axioms apply to a sequence of functors ${\displaystyle H_{n}}$ from the category of pairs (X, A) of topological spaces to the category of abelian groups, together with a natural transformation ${\displaystyle \partial :H_{i}(X,A)\to H_{i-1}(A)}$ called the boundary map (here H_i − 1(A) is a shorthand for H_i − 1(A,∅)). The axioms are:

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