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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 Poincaré complex ： ウィキペディア英語版 Poincaré complex In mathematics, and especially topology, a Poincaré complex (named after the mathematician Henri Poincaré) is an abstraction of the singular chain complex of a closed, orientable manifold. The singular homology and cohomology groups of a closed, orientable manifold are related by Poincaré duality. Poincaré duality is an isomorphism between homology and cohomology groups. A chain complex is called a Poincaré complex if its homology groups and cohomology groups have the abstract properties of Poincaré duality. A Poincaré space is a topological space whose singular chain complex is a Poincaré complex. These are used in surgery theory to analyze manifold algebraically. ==Definition== Let } be a chain complex, and assume that the homology groups of ''C'' are finitely generated. Assume that there exists a map , called a chain-diagonal, with the property that ; where the map denotes the ring homomorphism known as the augmentation map. It is defined as follows: if then Using the diagonal as defined above, we are able to form pairings, namely: :$\backslash rho\; :\; H^k(C)\backslash otimes\; H\_n(C)\; \backslash to\; H\_(C),\; \backslash \; \backslash text\; \backslash \; \backslash \; \backslash rho(x\backslash otimes\; y)\; =\; x\; \backslash frown\; y\; ,$ where $\backslash scriptstyle\; \backslash frown$ denotes the cap product. A chain complex ''C'' is called geometric if a chain-homotopy exists between Δ and τΔ, where is given by A geometric chain complex is called an algebraic Poincaré complex, of dimension ''n'', if there exists an infinite-ordered element of the ''n''-dimensional homology group, say , such that the maps given by :$(\backslash frown\backslash mu)\; :\; H^k(C)\; \backslash to\; H\_(C)$ are group isomorphisms for all . These isomorphisms are the isomorphisms of Poincaré duality. 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「Poincaré complex」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.