Eilenberg-Steenrod Axioms

A family of functors H_n(·) from the category of pairs of topological spaces and continuous maps, to the category of Abelian groups and group homomorphisms satisfies the Eilenberg-Steenrod axioms if the following conditions hold.

1. long exact sequence of a pair axiom. For every pair (X,A), there is a natural long exact sequence


where the map H_n(A)->H_n(X) is induced by the inclusion map A->X and H_n(X)->H_n(X,A) is induced by the inclusion map (X,phi)->(X,A). The map H_n(X,A)->H_(n-1)(A) is called the boundary map.

2. homotopy axiom. If f:(X,A)->(Y,B) is homotopic to g:(X,A)->(Y,B), then their induced maps f_*:H_n(X,A)->H_n(Y,B) and g_*:H_n(X,A)->H_n(Y,B) are the same.

3. excision axiom. If X is a space with subspaces A and U such that the set closure of U is contained in the interior of A, then the inclusion map (X\U,A\U)->(X,A) induces an isomorphism H_n(X U,A U)->H_n(X,A).

4. dimension axiom. Let X be a single point space. H_n(X)=0 unless n=0, in which case H_0(X)=G where G are some groups. The H_0 are called the coefficients of the homology theory H(·).

These are the axioms for a generalized homology theory. For a cohomology theory, instead of requiring that H(·) be a functor, it is required to be a co-functor (meaning the induced map points in the opposite direction). With that modification, the axioms are essentially the same (except that all the induced maps point backwards).

See also

Aleksandrov-Čech Cohomology

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Cite this as:

Weisstein, Eric W. "Eilenberg-Steenrod Axioms." From MathWorld--A Wolfram Web Resource.

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