|
|
| Line 1: |
Line 1: |
| 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]].
| | Nice to meet you, my name is Refugia. Supervising is my occupation. The favorite hobby for my children and me is to perform baseball but I haven't made a dime with it. Years in the past we moved to North Dakota and I love each day living right here.<br><br>Feel free to visit my web blog std testing at home ([http://tfor.vectorsigma.ru/node/5409 official statement]) |
| | |
| 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''<sub>''i'' − 1</sub>(''A'') is a shorthand for ''H''<sub>''i'' − 1</sub>(''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''<sub>0</sub>(''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]]
| |
Nice to meet you, my name is Refugia. Supervising is my occupation. The favorite hobby for my children and me is to perform baseball but I haven't made a dime with it. Years in the past we moved to North Dakota and I love each day living right here.
Feel free to visit my web blog std testing at home (official statement)