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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Eilenberg–Steenrod axioms ： ウィキペディア英語版 Eilenberg–Steenrod axioms In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces 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 functors 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.〔http://www.math.uiuc.edu/K-theory/0245/survey.pdf〕 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. ==Formal definition== The Eilenberg–Steenrod axioms apply to a sequence of functors $H\_n$ from the category of pairs (''X'', ''A'') of topological spaces to the category of abelian groups, together with a natural transformation $\backslash partial\; :\; H\_(X,\; A)\; \backslash to\; H\_(A)$ 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 $g:(X,\; A)\; \backslash rightarrow\; (Y,B)$ is homotopic to $h:(X,\; A)\; \backslash rightarrow\; (Y,B)$, 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 $i\; :\; (X-U,\; A-U)\; \backslash to\; (X,\; A)$ induces an isomorphism in homology. # Dimension: Let ''P'' be the one-point space; then $H\_n(P)\; =\; 0$ for all $n\; \backslash neq\; 0$. # Additivity: If $X\; =\; \backslash coprod\_$, then $H\_n(X)\; \backslash cong\; \backslash bigoplus\_\; H\_n(X\_).$ # Exactness: Each pair ''(X, A)'' induces a long exact sequence in homology, via the inclusions $i:\; A\; \backslash to\; X$ and $j:\; X\; \backslash to\; (X,\; A)$: ::$\backslash cdots\; \backslash to\; H\_n(A)\; \backslash to^\; H\_n(X)\; \backslash to^\; H\_n\; (X,A)\; \backslash to^\; H\_(A)\; \backslash to\; \backslash cdots.$ 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Eilenberg–Steenrod axioms」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.