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2013-07-08T04:57:06Z

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In [[mathematics]], specifically in [[algebraic topology]], the '''Eilenberg–Steenrod axioms''' are properties that [[homology theory|homology theories]] of [[topological space]]s 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 [[functor]]s 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.<ref>http://www.math.uiuc.edu/K-theory/0245/survey.pdf</ref>

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 theory|cobordism]].

==Formal definition==

The Eilenberg–Steenrod axioms apply to a sequence of functors <math>H_n</math> from the [[category (mathematics)|category]] of [[topological pair|pairs]] (''X'', ''A'') of topological spaces to the category of abelian [[group (mathematics)|group]]s, together with a [[natural transformation]] <math>\partial : H_{i}(X, A) \to H_{i-1}(A)</math> called the '''boundary map''' (here ''H''''i'' − 1(''A'') is a shorthand for ''H''''i'' − 1(''A'',∅)). The axioms are:

# '''Homotopy''': Homotopic maps induce the same map in homology. That is, if <math>g:(X, A) \rightarrow (Y,B)</math> is [[homotopic]] to <math>h:(X, A) \rightarrow (Y,B)</math>, then their induced [[Map (mathematics)|maps]] are the same.
# '''[[Excision theorem|Excision]]''': If (''X'', ''A'') is a pair and ''U'' is a subset of ''X'' such that the closure of ''U'' is contained in the interior of ''A'', then the inclusion map <math>i : (X-U, A-U) \to (X, A)</math> induces an [[isomorphism]] in homology.
# '''Dimension''': Let ''P'' be the one-point space; then <math>H_n(P) = 0</math> for all <math>n \neq 0</math>.
# '''Additivity''': If <math>X = \coprod_{\alpha}{X_{\alpha}}</math>, the disjoint union of a family of topological spaces <math>X_{\alpha}</math>, then <math>H_n(X) \cong \bigoplus_{\alpha} H_n(X_{\alpha}).</math>
# '''Exactness''': Each pair ''(X, A)'' induces a [[long exact sequence]] in homology, via the inclusions <math>i: A \to X</math> and <math>j: X \to (X, A)</math>:
::<math> \cdots \to H_n(A) \to^{\!\!\!\!\!\! i_*} H_n(X) \to^{\!\!\!\!\!\! j_*} H_n (X,A) \to^{\!\!\!\!\!\!\partial_*} H_{n-1}(A) \to \cdots.</math>

If ''P'' is the one point space then ''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.

==Consequences==

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''-[[sphere]]s, can be calculated directly from the axioms. From this it can be easily shown that the (''n'' − 1)-sphere is not a [[retract]] of the ''n''-disk. This is used in a proof of the [[Brouwer fixed point theorem]].

==Dimension axiom==

A "homology-like" theory satisfying all of the Eilenberg–Steenrod axioms except the dimension axiom is called an '''[[extraordinary homology theory]]''' (dually, '''[[extraordinary cohomology theory]]'''). Important examples of these were found in the 1950s, such as [[topological K-theory]] and [[cobordism theory]], which are extraordinary ''co''homology theories, and come with homology theories dual to them.

==See also==
* [[Zig-zag lemma]]

==References==
* Samuel Eilenberg, Norman E. Steenrod, ''Axiomatic approach to homology theory'', Proc. Nat. Acad. Sci. U. S. A. 31, (1945). 117–120.
* Samuel Eilenberg, Norman E. Steenrod, ''Foundations of algebraic topology'', [[Princeton University Press]], Princeton, New Jersey, 1952. xv+328 pp.
* [[Glen Bredon]]: ''Topology and Geometry'', 1993, ISBN 0-387-97926-3.

==Notes==
{{reflist}}

{{DEFAULTSORT:Eilenberg-Steenrod axioms}}
[[Category:Homology theory]]
[[Category:Mathematical axioms]]

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