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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 obstruction ： ウィキペディア英語版 Surgery obstruction In mathematics, specifically in surgery theory, the surgery obstructions define a map $\backslash theta\; \backslash colon\; \backslash mathcal\; (X)\; \backslash to\; L\_n\; (\backslash pi\_1\; (X))$ from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property when $n\; \backslash geq\; 5$: A degree-one normal map $(f,b)\; \backslash colon\; M\; \backslash to\; X$ is normally cobordant to a homotopy equivalence if and only if the image $\backslash theta\; (f,b)=0$ in $L\_n\; (\backslash mathbb\; ((X)))$. ==Sketch of the definition== The surgery obstruction of a degree-one normal map has a relatively complicated definition. Consider a degree-one normal map $(f,b)\; \backslash colon\; M\; \backslash to\; X$. The idea in deciding the question whether it is normally cobordant to a homotopy equivalence is to try to systematically improve $(f,b)$ so that the map $f$ becomes $m$-connected (that means the homotopy groups $\backslash pi\_$ * (f)=0 for $$ * \leq m) for high $m$. It is a consequence of Poincaré duality that if we can achieve this for $m\; >\; \backslash lfloor\; n/2\; \backslash rfloor$ then the map $f$ already is a homotopy equivalence. The word ''systematically'' above refers to the fact that one tries to do surgeries on $M$ to kill elements of $\backslash pi\_i\; (f)$. In fact it is more convenient to use homology of the universal covers to observe how connected the map $f$ is. More precisely, one works with the surgery kernels $K\_i\; (\backslash tilde\; M)\; :\; =\; \backslash mathrm\; \backslash$ which one views as $\backslash mathbb((X))$-modules. If all these vanish, then the map $f$ is a homotopy equivalence. As a consequence of Poincaré duality on $M$ and $X$ there is a $\backslash mathbb((X))$-modules Poincaré duality $K^\; (\backslash tilde\; M)\; \backslash cong\; K\_i\; (\backslash tilde\; M)$, so one only has to watch half of them, that means those for which $i\; \backslash leq\; \backslash lfloor\; n/2\; \backslash rfloor$. Any degree-one normal map can be made $\backslash lfloor\; n/2\; \backslash rfloor$-connected by the process called surgery below the middle dimension. This is the process of killing elements of $K\_i\; (\backslash tilde\; M)$ for described here when we have $p+q\; =\; n$ such that . After this is done there are two cases. 1. If $n=2k$ then the only nontrivial homology group is the kernel $K\_k\; (\backslash tilde\; M)\; :\; =\; \backslash mathrm\; \backslash$. It turns out that the cup-product pairings on $M$ and $X$ induce a cup-product pairing on $K\_k(\backslash tilde\; M)$. This defines a symmetric bilinear form in case $k=2l$ and a skew-symmetric bilinear form in case $k=2l+1$. It turns out that these forms can be refined to $\backslash varepsilon$-quadratic forms, where $\backslash varepsilon\; =\; (-1)^k$. These $\backslash varepsilon$-quadratic forms define elements in the L-groups $L\_n\; (\backslash pi\_1\; (X))$. 2. If $n=2k+1$ the definition is more complicated. Instead of a quadratic form one obtains from the geometry a quadratic formation, which is a kind of automorphism of quadratic forms. Such a thing defines an element in the odd-dimensional L-group $L\_n\; (\backslash pi\_1\; (X))$. If the element $\backslash theta\; (f,b)$ is zero in the L-group surgery can be done on $M$ to modify $f$ to a homotopy equivalence. Geometrically the reason why this is not always possible is that performing surgery in the middle dimension to kill an element in $K\_k\; (\backslash tilde\; M)$ possibly creates an element in $K\_\; (\backslash tilde\; M)$ when $n\; =\; 2k$ or in $K\_\; (\backslash tilde\; M)$ when $n=2k+1$. So this possibly destroys what has already been achieved. However, if $\backslash theta\; (f,b)$ is zero, surgeries can be arranged in such a way that this does not happen. 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 obstruction」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.