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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 Size homotopy group ： ウィキペディア英語版 Size homotopy group The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair $(M,\backslash varphi)\backslash$ is given, where $M\backslash$ is a closed manifold of class $C^0\backslash$ and $\backslash varphi:M\backslash to\; \backslash mathbb^k\backslash$ is a continuous function. Let us consider the partial order $\backslash preceq\backslash$ in $\backslash mathbb^k\backslash$ defined by setting $(x\_1,\backslash ldots,x\_k)\backslash preceq(y\_1,\backslash ldots,y\_k)\backslash$ if and only if $x\_1\; \backslash le\; y\_1,\backslash ldots,\; x\_k\; \backslash le\; y\_k\backslash$. For every $Y\backslash in\backslash mathbb^k\backslash$ we set $M\_=\backslash \backslash$. Assume that $P\backslash in\; M\_X\backslash$ and $X\backslash preceq\; Y\backslash$. If $\backslash alpha\backslash$, $\backslash beta\backslash$ are two paths from $P\backslash$ to $P\backslash$ and a homotopy from $\backslash alpha\backslash$ to $\backslash beta\backslash$, based at $P\backslash$, exists in the topological space $M\_\backslash$, then we write $\backslash alpha\; \backslash approx\_\backslash beta\backslash$. The first size homotopy group of the size pair $(M,\backslash varphi)\backslash$ computed at $(X,Y)\backslash$ is defined to be the quotient set of the set of all paths from $P\backslash$ to $P\backslash$ in $M\_X\backslash$ with respect to the equivalence relation $\backslash approx\_\backslash$, endowed with the operation induced by the usual composition of based loops.〔Patrizio Frosini, Michele Mulazzani, ''Size homotopy groups for computation of natural size distances'', Bulletin of the Belgian Mathematical Society – Simon Stevin, 6:455–464, 1999.〕 In other words, the first size homotopy group of the size pair $(M,\backslash varphi)\backslash$ computed at $(X,Y)\backslash$ and $P\backslash$ is the image $h\_(\backslash pi\_1(M\_X,P))\backslash$ of the first homotopy group $\backslash pi\_1(M\_X,P)\backslash$ with base point $P\backslash$ of the topological space $M\_X\backslash$, when $h\_\backslash$ is the homomorphism induced by the inclusion of $M\_X\backslash$ in $M\_Y\backslash$. The $n\backslash$-th size homotopy group is obtained by substituting the loops based at $P\backslash$ with the continuous functions $\backslash alpha:S^n\backslash to\; M\backslash$ taking a fixed point of $S^n\backslash$ to $P\backslash$, as happens when higher homotopy groups are defined. ==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「Size homotopy group」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.