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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 cup product ： ウィキペディア英語版 cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space ''X'' into a graded ring, ''H''∗(''X''), called the cohomology ring. The cup product was introduced in work of J. W. Alexander, Eduard Čech and Hassler Whitney from 1935–1938, and, in full generality, by Samuel Eilenberg in 1944. ==Definition== In singular cohomology, the cup product is a construction giving a product on the graded cohomology ring ''H''∗(''X'') of a topological space ''X''. The construction starts with a product of cochains: if ''c''''p'' is a ''p''-cochain and ''d''''q'' is a ''q''-cochain, then :$(c^p\; \backslash smile\; d^q)(\backslash sigma)\; =\; c^p(\backslash sigma\; \backslash circ\; \backslash iota\_)\; \backslash cdot\; d^q(\backslash sigma\; \backslash circ\; \backslash iota\_)$ where σ is a singular (''p'' + ''q'') -simplex and $\backslash iota\_S\; ,\; S\; \backslash subset\; \backslash$ is the canonical embedding of the simplex spanned by S into the $(p+q)$-simplex whose vertices are indexed by $\backslash$. Informally, $\backslash sigma\; \backslash circ\; \backslash iota\_$ is the ''p''-th front face and $\backslash sigma\; \backslash circ\; \backslash iota\_$ is the ''q''-th back face of σ, respectively. The coboundary of the cup product of cocycles c''p'' and d''q'' is given by :$\backslash delta(c^p\; \backslash smile\; d^q)\; =\; \backslash delta\; \backslash smile\; d^q\; +\; (-1)^p(c^p\; \backslash smile\; \backslash delta).$ The cup product of two cocycles is again a cocycle, and the product of a coboundary with a cocycle (in either order) is a coboundary. The cup product operation induces a bilinear operation on cohomology, : $H^p(X)\; \backslash times\; H^q(X)\; \backslash to\; H^(X).$ 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「cup product」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.