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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 Hopf invariant ： ウィキペディア英語版 Hopf invariant In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres. == Motivation == In 1931 Heinz Hopf used Clifford parallels to construct the ''Hopf map'' :$\backslash eta\backslash colon\; S^3\; \backslash to\; S^2$, and proved that $\backslash eta$ is essential, i.e. not homotopic to the constant map, by using the linking number (=1) of the circles :$\backslash eta^(x),\backslash eta^(y)\; \backslash subset\; S^3$ for any $x\; \backslash neq\; y\; \backslash in\; S^2$. It was later shown that the homotopy group $\backslash pi\_3(S^2)$ is the infinite cyclic group generated by $\backslash eta$. In 1951, Jean-Pierre Serre proved that the rational homotopy groups :$\backslash pi\_i(S^n)\; \backslash otimes\; \backslash mathbb$ for an odd-dimensional sphere ($n$ odd) are zero unless ''i'' = 0 or ''n''. However, for an even-dimensional sphere (''n'' even), there is one more bit of infinite cyclic homotopy in degree $2n-1$. 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「Hopf invariant」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.