Eilenberg–Steenrod axioms

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.

Formal definition

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

1. Homotopy: Homotopic maps induce the same map in homology. That is, if ${\displaystyle g:(X,A)\to (Y,B)}$ is homotopic to ${\displaystyle h:(X,A)\to (Y,B)}$, then their induced homomorphisms are the same.
2. Excision: If ${\displaystyle (X,A)}$ 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 ${\displaystyle i:(X\setminus U,A\setminus U)\to (X,A)}$ induces an isomorphism in homology.
3. Dimension: Let P be the one-point space; then ${\displaystyle H_n(P)=0}$ for all ${\displaystyle n\ne 0}$.
4. Additivity: If ${\displaystyle X=\coprod _{\alpha }{X}_{\alpha }}$, the disjoint union of a family of topological spaces ${\displaystyle X_{\alpha }}$, then ${\displaystyle H_n(X)\cong \underset{\alpha }{⨁}{H}_n(X_{\alpha }).}$
5. Exactness: Each pair (X, A) induces a long exact sequence in homology, via the inclusions ${\displaystyle i:A\to X}$ and ${\displaystyle j:X\to (X,A)}$:

${\displaystyle \cdots \to H_n(A)\stackrel{i_{\ast }}{\to }{H}_n(X)\stackrel{j_{\ast }}{\to }{H}_n(X,A)\stackrel{\mathrm{\partial }}{\to }{H}_{n-1}(A)\to \cdots .}$

If P is the one point space, then ${\displaystyle H_0(P)}$ is called the coefficient group. For example, singular homology (taken with integer coefficients, as is most common) has as coefficients the integers.