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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 Serre spectral sequence ： ウィキペディア英語版 Serre spectral sequence In mathematics, the Serre spectral sequence (sometimes Leray-Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological algebra the singular (co)homology of the total space ''X'' of a (Serre) fibration in terms of the (co)homology of the base space ''B'' and the fiber ''F''. The result is due to Jean-Pierre Serre in his doctoral dissertation. ==Cohomology spectral sequence== Let ''f'' : ''X'' → ''B'' be a Serre fibration of topological spaces, and let ''F'' be the fiber. The Serre cohomology spectral sequence is the following: :$E\_2^\; =\; H^p(B,\; H^q(F))\; \backslash Rightarrow\; H^(X).$ Here, at least under standard simplifying conditions, the coefficient group in the ''E''2-term is the ''q''-th integral cohomology group of ''F'', and the outer group is the singular cohomology of ''B'' with coefficients in that group. Strictly speaking, what is meant is cohomology with respect to the local coefficient system on ''B'' given by the cohomology of the various fibers. Assuming for example, that ''B'' is simply connected, this collapses to the usual cohomology. For a path connected base, all the different fibers are homotopy equivalent. In particular, their cohomology is isomorphic, so the choice of "the" fiber does not give any ambiguity. The abutment means integral cohomology of the total space ''X''. This spectral sequence can be derived from an exact couple built out of the long exact sequences of the cohomology of the pair (''Xp'', ''X''''p''−1), where ''Xp'' is the restriction of the fibration over the ''p''-skeleton of ''B''. More precisely, using this notation, : $A\; =\; \backslash bigoplus\_\; H^q(X\_p)$, $E\_1^\; =\; C\; =\; \backslash bigoplus\_\; H^q(X\_p,\; X\_),$ ''f'' is defined by restricting each piece on ''Xp'' to ''X''''p''−1, ''g'' is defined using the coboundary map in the long exact sequence of the pair, and ''h'' is defined by restricting (''Xp'', ''X''''p''−1) to ''Xp''. There is a multiplicative structure :$E\_r^\; \backslash times\; E\_r^\; \backslash to\; E\_r^,$ coinciding on the ''E''2-term with (−1)''qs'' times the cup product, and with respect to which the differentials ''d''r are (graded) derivations inducing the product on the ''E''''r''+1-page from the one on the ''Er''-page. 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「Serre spectral sequence」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.