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In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Originally developed for differentiable (= smooth) manifolds, surgery techniques also apply to PL (= piecewise linear) and topological manifolds. Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. It is a major tool in the study and classification of manifolds of dimension greater than 3. More technically, the idea is to start with a well-understood manifold ''M'' and perform surgery on it to produce a manifold ''M ''′ having some desired property, in such a way that the effects on the homology, homotopy groups, or other interesting invariants of the manifold are known. The classification of exotic spheres by led to the emergence of surgery theory as a major tool in high-dimensional topology. ==Surgery on a manifold== Recall that in general, if ''X'', ''Y'' are manifolds with boundary, then the boundary of the product manifold is ∂(''X'' × ''Y'')=(∂''X'' × ''Y'') ∪ (''X'' × ∂''Y''). The basic observation which justifies surgery is that the space S''p'' × S''q''-1 can be understood either as the boundary of D''p''+1 × S''q''-1 or as the boundary of S''p'' × D''q''. In symbols, ∂(S''p'' × D''q'') = S''p'' × S''q''-1 = ∂(D''p''+1 × S''q''-1), where D''q'' is the q-dimensional disk, i.e., the set of q-dimensional points that are at distance one-or-less from a given fixed point (the center of the disk); for example, then, D''1'' is (equivalent, or homeomorphic to), the unit interval, while D''2'' is a circle together with the points in its interior. Now, given a manifold ''M'' of dimension ''n'' = ''p+q'' and an embedding : S''p'' × D''q'' → ''M'', define another ''n''-dimensional manifold ''M''′ to be : One says that the manifold ''M''′ is produced by a ''surgery'' cutting out S''p'' × D''q'' and gluing in D''p''+1 × S''q''-1, or by a ''p''-''surgery'' if one wants to specify the number ''p''. Strictly speaking, ''M''′ is a manifold with corners, but there is a canonical way to smooth them out. Notice that the submanifold that was replaced in ''M'' was of the same dimension as ''M'' (it was of codimension 0). Surgery is closely related to (but not the same as) handle attaching. Given an (''n''+1)-manifold with boundary (''L'', ∂''L'') and an embedding : S''p'' × D''q'' → ∂''L'', where ''n'' = ''p+q'', define another (''n''+1)-manifold with boundary ''L''′ by : The manifold ''L''′ is obtained by ''attaching a (''p''+1)-handle'', with ∂''L''′ obtained from ∂''L'' by a ''p''-surgery : A surgery on ''M'' not only produces a new manifold ''M''′, but also a cobordism ''W'' between ''M'' and ''M''′. The ''trace'' of the surgery is the cobordism (''W''; ''M'', ''M''′), with : the (''n''+1)-dimensional manifold with boundary ∂''W'' = ''M'' ∪ ''M''′ obtained from the product ''M'' × ''I'' by attaching a (''p''+1)-handle D''p''+1 × D''q''. Surgery is symmetric in the sense that the manifold ''M'' can be re-obtained from ''M''′ by a (''q''-1)-surgery, the trace of which coincides with the trace of the original surgery, up to orientation. In most applications, the manifold ''M'' comes with additional geometric structure, such as a map to some reference space, or additional bundle data. One then wants the surgery process to endow ''M''′ with the same kind of additional structure. For instance, a standard tool in surgery theory is surgery on normal maps: such a process changes a normal map to another normal map within the same bordism class. 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 theory」の詳細全文を読む スポンサード リンク
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