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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 Sphere theorem (3-manifolds) ： ウィキペディア英語版 Sphere theorem (3-manifolds) In mathematics, in the topology of 3-manifolds, the sphere theorem of gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres. One example is the following: Let $M$ be an orientable 3-manifold such that $\backslash pi\_2(M)$ is not the trivial group. Then there exists a non-zero element of $\backslash pi\_2(M)$ having a representative that is an embedding $S^2\backslash to\; M$. The proof of this version can be based on transversality methods, see Batude below. Another more general version (also called the projective plane theorem due to Epstein) is: Let $M$ be any 3-manifold and $N$ a $\backslash pi\_1(M)$-invariant subgroup of $\backslash pi\_2(M)$. If $f\backslash colon\; S^2\backslash to\; M$ is a general position map such that $()\backslash notin\; N$ and $U$ is any neighborhood of the singular set $\backslash Sigma(f)$, then there is a map $g\backslash colon\; S^2\backslash to\; M$ satisfying #$()\backslash notin\; N$, #$g(S^2)\backslash subset\; f(S^2)\backslash cup\; U$, #$g\backslash colon\; S^2\backslash to\; g(S^2)$ is a covering map, and #$g(S^2)$ is a 2-sided submanifold (2-sphere or projective plane) of $M$. quoted in Hempel (p. 54) ==References== * * * * * 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「Sphere theorem (3-manifolds)」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.