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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 Surgery exact sequence ： ウィキペディア英語版 Surgery exact sequence In the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension $>4$. The surgery structure set $\backslash mathcal\; (X)$ of a compact $n$-dimensional manifold $X$ is a pointed set which classifies $n$-dimensional manifolds within the homotopy type of $X$. The basic idea is that in order to calculate $\backslash mathcal\; (X)$ it is enough to understand the other terms in the sequence, which are usually easier to determine. These are on one hand the normal invariants which form generalized cohomology groups, and hence one can use standard tools of algebraic topology to calculate them at least in principle. On the other hand there are the L-groups which are defined algebraically in terms of quadratic forms or in terms of chain complexes with quadratic structure. A great deal is known about these groups. Another part of the sequence are the surgery obstruction maps from normal invariants to the L-groups. For these maps there are certain characteristic classes formulas, which enable to calculate them in some cases. Knowledge of these three components, that means: the normal maps, the L-groups and the surgery obstruction maps is enough to determine the structure set (at least up to extension problems). In practice one has to proceed case by case, for each manifold $\backslash mathcal\{\}\; X$ it is a unique task to determine the surgery exact sequence, see some examples below. Also note that there are versions of the surgery exact sequence depending on the category of manifolds we work with: smooth (DIFF), PL, or topological manifolds and whether we take Whitehead torsion into account or not (decorations $s$ or $h$). The original 1962 work of Browder and Novikov on the existence and uniqueness of manifolds within a simply-connected homotopy type was reformulated by Sullivan in 1966 as a surgery exact sequence. In 1970 Wall developed non-simply-connected surgery theory and the surgery exact sequence for manifolds with arbitrary fundamental group. ==Definition== The surgery exact sequence is defined as : $$ \cdots \to \mathcal_\partial (X \times I) \to L_ (\pi_1 (X)) \to \mathcal(X) \to \mathcal (X) \to L_n (\pi_1 (X)) where: the entries $\backslash mathcal\_\backslash partial\; (X\; \backslash times\; I)$ and $\backslash mathcal\; (X)$ are the abelian groups of normal invariants, the entries $\backslash mathcal\; L\_\; (\backslash pi\_1\; (X))$ are the L-groups associated to the group ring $\backslash mathbb((X))$, the maps $\backslash theta\; \backslash colon\; \backslash mathcal\_\backslash partial\; (X\; \backslash times\; I)\; \backslash to\; L\_\; (\backslash pi\_1\; (X))$ and $\backslash theta\; \backslash colon\; \backslash mathcal\; (X)\; \backslash to\; L\_n\; (\backslash pi\_1\; (X))$ are the surgery obstruction maps, the arrows $\backslash partial\; \backslash colon\; L\_\; (\backslash pi\_1\; (X))\; \backslash to\; \backslash mathcal(X)$ and $\backslash eta\; \backslash colon\; \backslash mathcal(X)\; \backslash to\; \backslash mathcal\; (X)$ will be explained below. 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「Surgery exact sequence」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.