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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 Seifert–van Kampen theorem ： ウィキペディア英語版 Seifert–van Kampen theorem In mathematics, the Seifert–van Kampen theorem of algebraic topology, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space $X$, in terms of the fundamental groups of two open, path-connected subspaces $U$ and $V$ that cover $X$. It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones. The underlying idea is that paths in $X$ can be partitioned into journeys: through the intersection $W$ of $U$ and $V$, through $U$ but outside $V$, and through $V$ outside $U$. In order to move segments of paths around, by homotopy to form loops returning to a base point $w$ in $W$, we should assume $U$, $V$ and $W$ are path-connected and that $W$ is not empty. We also assume that $U$ and $V$ are open subspaces with union $X$. == Equivalent formulations == In the language of combinatorial group theory, $\backslash pi\_1(X,w)$ is the free product with amalgamation of $\backslash pi\_1(U,w)$ and $\backslash pi\_1(V,w)$, with respect to the (not necessarily injective) homomorphisms $I$ and $J$. Given group presentations: :$\backslash pi\_1(U,w)\; =\; \backslash langle\; u\_1,\backslash ldots,u\_k\; \backslash mid\backslash alpha\_1,\backslash ldots,\backslash alpha\_l\backslash rangle$ :$\backslash pi\_1(V,w)\; =\; \backslash langle\; v\_1,\backslash ldots,v\_m\; \backslash mid\; \backslash beta\_1,\backslash ldots,\backslash beta\_n\backslash rangle,$ and :$\backslash pi\_1(W,w)\; =\; \backslash langle\; w\_1,\backslash ldots,w\_p\; \backslash mid\; \backslash gamma\_1,\backslash ldots,\backslash gamma\_q\backslash rangle$ the amalgamation can be presented as :$\backslash pi\_1(X,w)\; =\; \backslash langle\; u\_1,\backslash ldots,u\_k,\; v\_1,\backslash ldots,v\_m\; \backslash mid\; \backslash alpha\_1,\backslash ldots,\backslash alpha\_l,\; \backslash beta\_1,\backslash ldots,\backslash beta\_n,\; I(w\_1)J(w\_1)^,\backslash ldots,I(w\_p)J(w\_p)^\backslash rangle.$ In category theory, $\backslash pi\_1(X,w)$ is the pushout, in the category of groups, of the diagram: :$\backslash pi\_1(U,w)\backslash gets\backslash pi\_1(W,w)\backslash to\backslash pi\_1(V,w).$ 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「Seifert–van Kampen 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.