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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 Loop theorem ： ウィキペディア英語版 Loop theorem In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem. A simple and useful version of the loop theorem states that if there is a map :$f\backslash colon\; (D^2,\backslash partial\; D^2)\backslash to\; (M,\backslash partial\; M)\; \backslash ,$ with $f|\backslash partial\; D^2$ not nullhomotopic in $\backslash partial\; M$, then there is an embedding with the same property. The following version of the loop theorem, due to John Stallings, is given in the standard 3-manifold treatises (such as Hempel or Jaco): Let $M$ be a 3-manifold and let $S$ be a connected surface in $\backslash partial\; M$. Let $N\backslash subset$ \pi_1(S) be a normal subgroup such that $\backslash mathop\{\backslash mathrm\{ker\}\}(\backslash pi\_1(S)\; \backslash to\; \backslash pi\_1(M))\; -\; N\; \backslash neq\; \backslash emptyset$. Let :$f\; \backslash colon\; D^2\backslash to\; M\; \backslash ,$ be a continuous map such that :$f(\backslash partial\; D^2)\backslash subset\; S\; \backslash ,$ and :$(D^2)\backslash notin\; N.\; \backslash ,$ Then there exists an embedding :$g\backslash colon\; D^2\backslash to\; M\; \backslash ,$ such that :$g(\backslash partial\; D^2)\backslash subset\; S\; \backslash ,$ and :$(D^2)\backslash notin\; N.\; \backslash ,$ Furthermore if one starts with a map ''f'' in general position, then for any neighborhood U of the singularity set of ''f'', we can find such a ''g'' with image lying inside the union of image of ''f'' and U. Stalling's proof utilizes an adaptation, due to Whitehead and Shapiro, of Papakyriakopoulos' "tower construction". The "tower" refers to a special sequence of coverings designed to simplify lifts of the given map. The same tower construction was used by Papakyriakopoulos to prove the sphere theorem (3-manifolds), which states that a nontrivial map of a sphere into a 3-manifold implies the existence of a nontrivial ''embedding'' of a sphere. There is also a version of Dehn's lemma for minimal discs due to Meeks and S.-T. Yau, which also crucially relies on the tower construction. A proof not utilizing the tower construction exists of the first version of the loop theorem. This was essentially done 30 years ago by Friedhelm Waldhausen as part of his solution to the word problem for Haken manifolds; although he recognized this gave a proof of the loop theorem, he did not write up a detailed proof. The essential ingredient of this proof is the concept of Haken hierarchy. Proofs were later written up, by Klaus Johannson, Marc Lackenby, and Iain Aitchison with Hyam Rubinstein. ==References== *W. Jaco, ''Lectures on 3-manifolds topology'', A.M.S. regional conference series in Math 43. *J. Hempel, ''3-manifolds'', Princeton University Press 1976. * Hatcher, ''Notes on basic 3-manifold topology'', (available online ) Category:3-manifolds Category:Continuous mappings Category:Theorems in topology 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「Loop 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.