翻訳と辞書 Words near each other ・ Loop modeling ・ Loop nest optimization ・ Loop of Henle ・ Loop optimization ・ Loop Parkway ・ Loop performance ・ Loop quantum cosmology ・ Loop quantum gravity ・ Loop recorder ・ Loop recording ・ Loop representation in gauge theories and quantum gravity ・ Loop Retail Historic District ・ Loop rock ・ Loop scheduling ・ Loop Service (Portland Streetcar) ・ Loop space ・ Loop splitting ・ Loop start ・ Loop subdivision surface ・ Loop the Loop (Coney Island) ・ Loop the Loop (Olentangy Park) ・ Loop the Loop (Young's Million Dollar Pier) ・ Loop theorem ・ Loop tiling ・ Loop unrolling ・ Loop unswitching ・ Loop v. Litchfield ・ Loop variant ・ Loop, Germany ・ Loop, Texas Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Loop space ： ウィキペディア英語版 Loop space In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of based maps from the circle ''S''1 to ''X'' with the compact-open topology. Two elements of a loop space can be naturally concatenated. With this concatenation operation, a loop space is an ''A''∞-space. The adjective ''A''∞ describes the manner in which concatenating loops is homotopy coherently associative. The quotient of the loop space Ω''X'' by the equivalence relation of pointed homotopy is the fundamental group ''π''1(''X''). The iterated loop spaces of ''X'' are formed by applying Ω a number of times. An analogous construction of topological spaces without basepoint is the free loop space. The free loop space of a topological space ''X'' is the space of maps from ''S''1 to ''X'' with the compact-open topology. That is to say, the free loop space of a topological space ''X'' is the function space $\backslash mathrm(S^1,X)$. The free loop space of ''X'' is denoted by $\backslash mathcalX$. The free loop space construction is right adjoint to the cartesian product with the circle, while the loop space construction is right adjoint to the reduced suspension. This adjunction accounts for much of the importance of loop spaces in stable homotopy theory. == Relation between homotopy groups of a space and those of its loop space == The basic relation between the homotopy groups is $\backslash pi\_k(X)\; \backslash approxeq\; \backslash pi\_(\backslash Omega\; X)$.〔http://topospaces.subwiki.org/wiki/Loop_space_of_a_based_topological_space〕 More generally, :$(Z,X)\; \backslash approxeq\; (\backslash Omega\; X)$ where, $()$ is the set of homotopy classes of maps $A\; \backslash rightarrow\; B$, and $\backslash Sigma\; A$ is the suspension of A. In general $(B)$ does not have a group structure for arbitrary spaces $A$ and $B$. However, it can be shown that $(Z,X)$ and $(\backslash Omega\; X)$ do have natural group structures when $Z$ and $X$ are pointed, and the aforesaid isomorphism is of those groups. 〔 (chapter 8, section 2)〕 Note that setting $Z\; =\; S^$ (the $k-1$ sphere) gives the earlier result. 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 space」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.