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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–Zilber theorem ： ウィキペディア英語版 Eilenberg–Zilber theorem In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space $X\; \backslash times\; Y$ and those of the spaces $X$ and $Y$. The theorem first appeared in a 1953 paper in the American Journal of Mathematics. ==Statement of the theorem== The theorem can be formulated as follows. Suppose $X$ and $Y$ are topological spaces, Then we have the three chain complexes $C\_$ *(X), $C\_$ *(Y), and $C\_$ *(X \times Y) . (The argument applies equally to the simplicial or singular chain complexes.) We also have the ''tensor product complex'' $C\_$ *(X) \otimes C_ *(Y), whose differential is, by definition, :$\backslash delta(\; \backslash sigma\; \backslash otimes\; \backslash tau)\; =\; \backslash delta\_X\; \backslash sigma\; \backslash otimes\; \backslash tau\; +\; (-1)^p\; \backslash sigma\; \backslash otimes\; \backslash delta\_Y\; \backslash tau$ for $\backslash sigma\; \backslash in\; C\_p(X)$ and $\backslash delta\_X$, $\backslash delta\_Y$ the differentials on $C\_$ *(X),$C\_$ *(Y). Then the theorem says that we have chain maps :$F:\; C\_$ *(X \times Y) \rightarrow C_ *(X) \otimes C_ *(Y), \quad G: C_ *(X) \otimes C_ *(Y) \rightarrow C_ *(X \times Y) such that $FG$ is the identity and $GF$ is chain-homotopic to the identity. Moreover, the maps are natural in $X$ and $Y$. Consequently the two complexes must have the same homology: :$H\_$ *(C_ *(X \times Y)) \cong H_ *(C_ *(X) \otimes C_ *(Y)). An important generalisation to the non-abelian case using crossed complexes is given in the paper by Tonks below. This give full details of a result on the (simplicial) classifying space of a crossed complex stated but not proved in the paper by Brown and Higgins on classifying spaces. 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–Zilber theorem」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.