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| Formal definition | |
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The Eilenberg–Steenrod axioms apply to a sequence of functors Hn from the category of pairs (X, A) of topological spaces to the category of abelian groups, together with a natural transformation called the boundary map (here Hi − 1(A) is a shorthand for Hi − 1(A,∅)). The axioms are:
Homotopy: Homotopic maps induce the same map in homology. That is, if is homotopic to , then their induced maps are the same.
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 induces an isomorphism in homology.
Dimension: Let P be the one-point space; then Hn(P) = 0 for all .
Additivity: If , the disjoint union of a family of topological spaces Xα, then
Exactness: Each pair (X, A) induces a long exact sequence in homology, via the inclusions and :
If P is the one point space then H0(P) is called the coefficient group. For example, singular homology (taken with integer coefficients, as is most common) has as coefficients the integers.
[edit] Tags:Topological Spaces,Singular Homology,Sequence,Functors,Edit,Category,Groups,Maps, | |
| Consequences | |
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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-spheres, 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 the proof of the Brouwer fixed point theorem.
[edit] Tags:Retract,Brouwer Fixed Point Theorem, | |
| Dimension axiom | |
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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 cohomology theories, and come with homology theories dual to them.
[edit] Tags:Homology Theories,Extraordinary Homology Theory,K-theory,Cobordism,Extraordinary Cohomology Theory,Topological K-theory,Cobordism Theory,Homology Theory, | |
| References | |
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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.
[edit] Tags:Algebraic Topology,Samuel Eilenberg,Princeton University Press,Glen Bredon,Isbn 0-387-97926-3, | |
| Notes | |
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^ http://www.math.uiuc.edu/K-theory/0245/survey.pdf
Retrieved from "http://en.wikipedia.org/w/index.php?title=Eilenberg%E2%80%93Steenrod_axioms&oldid=462048308"
Categories: Homology theoryMathematical axioms
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