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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 Assembly map ： ウィキペディア英語版 Assembly map In mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data. Assembly maps for algebraic K-theory and L-theory play a central role in the topology of high-dimensional manifolds, since their homotopy fibers have a direct geometric interpretation. Equivariant assembly maps are used to formulate the Farrell–Jones conjectures in K- and L-theory. ==Homotopy-theoretical viewpoint== It is a classical result that for any generalized homology theory $h\_$ * on the category of topological spaces (assumed to be homotopy equivalent to CW-complexes), there is a spectrum $E$ such that :$h\_$ *(X)\cong \pi_ *(X_+\wedge E), where $X\_+:=X\backslash coprod\; \backslash$. The functor $X\backslash mapsto\; X\_+\; \backslash wedge\; E$ from spaces to spectra has the following properties: * It is homotopy-invariant (preserves homotopy equivalences). This reflects the fact that $h\_$ * is homotopy-invariant. * It preserves homotopy co-cartesian squares. This reflects that fact that $h\_$ * has Mayer-Vietoris sequences, an equivalent characterization of excision. * It preserves arbitrary coproducts. This reflects the disjoint-union axiom of $h\_$ *. A functor from spaces to spectra fulfilling these properties is called excisive. Now suppose that $F$ is a homotopy-invariant, not necessarily excisive functor. An assembly map is a natural transformation $\backslash alpha\backslash colon\; F^\backslash \%\backslash to\; F$ from some excisive functor $F^\backslash \%$ to $F$ such that $F^\backslash \%($ *)\to F( *) is a homotopy equivalence. If we denote by $h\_$ *:=\pi_ *\circ F^\% the associated homology theory, it follows that the induced natural transformation of graded abelian groups $h\_$ *\to \pi_ *\circ F is the universal transformation from a homology theory to $\backslash pi\_$ *\circ F, i.e. any other transformation $k\_$ *\to\pi_ *\circ F from some homology theory $k\_$ * factors uniquely through a transformation of homology theories $k\_$ *\to h_ *. Assembly maps exist for any homotopy invariant functor, by a simple homotopy-theoretical construction. 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「Assembly map」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.