Steenrod-Eilenberg axioms

Axioms describing the basic properties of homology (cohomology) groups (cf. Cohomology group; Homology group), which uniquely define the relevant homology (cohomology) theory. An axiomatic homology theory is defined on a certain category of pairs $ ( X, A) $ of topological spaces if for any integer $ q $ an Abelian group (or module over some ring) $ H _ {q} ( X, A) $ is assigned to every pair $ ( X, A) $, while a homomorphism $ f _ \star : H _ {q} ( X, A) \rightarrow H _ {q} ( Y, B) $ is assigned to each mapping $ f: ( X, A) \rightarrow ( Y, B) $ in such a way that the following axioms are satisfied:

1) $ f _ \star $ is the identity isomorphism if $ f $ is the identity homeomorphism;

2) $ ( gf ) _ \star = g _ \star f _ \star $, where $ g: ( Y, B) \rightarrow ( Z, C) $;

3) connecting homomorphisms $ \partial : H _ {q} ( X, A) \rightarrow H _ {q-1} ( A) $ are defined such that $ \partial f _ \star = f _ \star \partial $( here $ A=( A, \emptyset) $, $ \emptyset $ is the empty set, while the mapping $ A \rightarrow B $, induced by $ f $, is also denoted by $ f $);

4) the exactness axiom: The homology sequence

$$ {} \dots \rightarrow H _ {q+1} ( X, A) \mathop \rightarrow \limits ^ \partial H _ {q} ( A) \rightarrow ^ { {i _ \star} } H _ {q} ( X) \rightarrow ^ { {j _ \star} } $$

$$ \rightarrow ^ { {j _ \star} } H _ {q} ( X, A) \mathop \rightarrow \limits ^ \partial H _ {q-1} ( A) \rightarrow \dots , $$

where $ i: A \subset X $, $ j: X \subset ( X, A) $ are inclusions, is exact, i.e. the kernel of every homomorphism coincides with the image of the previous one;

5) the homotopy axiom: $ f _ \star = f _ \star ^ { \prime } $ for homotopic mappings $ f, f ^ { \prime } : ( X, A) \rightarrow ( Y, B) $ in the category under consideration;

6) the excision axiom: If the closure in $ X $ of an open subset $ U $ in $ X $ is contained in the interior of $ A $, and the inclusion $ i: ( X\setminus U, A\setminus U) \subset ( X, A) $ belongs to the category, then $ i _ \star $ is an isomorphism;

7) the dimension axiom: $ H _ {q} ( P) = 0 $ when $ q \neq 0 $ for any singleton $ P $. The group $ H _ {0} ( P) $ is usually called the coefficient group. Axiomatic cohomology theories are dually defined (homomorphisms $ f ^ { \star } : H ^ {q} ( Y, B) \rightarrow H ^ {q} ( X, A) $ are assigned to mappings $ f $; the connecting homomorphisms take the form $ \delta : H ^ {q} ( A) \rightarrow H ^ {q+1} ( X, A) $). In the category of compact polyhedra, the ordinary homology and cohomology theories are the unique axiomatic theories with a given coefficient group (the uniqueness theorem). In the category of all polyhedra, the uniqueness theorem holds when the requirement is added that the homology (cohomology) of a union of open-closed, pairwise-disjoint subspaces be naturally isomorphic to the direct sum of the homology (direct product of the cohomology) of the subspaces (Milnor's additivity axiom). An axiomatic description of homology and cohomology theory also exists in more general categories of topological spaces (see [2], [3]). Generalized cohomology theories satisfy all the Steenrod–Eilenberg axioms (except for the dimension axiom), but are not uniquely defined by them.

References

[1]	S. Eilenberg, N.E. Steenrod, "Foundations of algebraic topology" , Princeton Univ. Press (1966)
[2]	S.V. Petkova, "On the axioms of homology theory" Math. USSR Sb. , 19 : 4 (1973) pp. 597–614 Mat. Sb. , 90 : 4 (1973) pp. 607–624
[3]	W.S. Massey, "Notes on homology and cohomology theory" , Yale Univ. Press (1964)

Comments

In the West, these axioms are invariably called the Eilenberg–Steenrod axioms; the reversal of the two names in Russian is a consequence of Cyrillic alphabetical order. Often the phrase "Eilenberg–Steenrod axioms" is used to refer only to the last four of the axioms above, the first three being taken as part of the definition of the functor to which the axioms apply. The four named axioms are independent; however, the homotopy axiom becomes redundant if one strengthens the dimension axiom to the assertion that the homology of any contractible space is the same as that of a point. It is also possible [a1] to axiomatize homology theories defined on categories whose objects are single spaces rather than pairs of spaces; in this formulation the exactness axiom is replaced by a Mayer–Vietoris axiom.

References

[a1]	G.M. Kelly, "Single-space axioms for homology theory" Proc. Cambridge Philos. Soc. , 55 (1959) pp. 10–22