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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 Kervaire invariant ： ウィキペディア英語版 Kervaire invariant In mathematics, the Kervaire invariant, named for Michel Kervaire, is defined in geometric topology. It is an invariant of a (4''k''+2)-dimensional (singly even-dimensional) framed differentiable manifold (or more generally PL-manifold) ''M,'' taking values in the 2-element group Z/2Z = . The Kervaire invariant is defined as the Arf invariant of the skew-quadratic form on the middle dimensional homology group. It can be thought of as the simply-connected ''quadratic'' L-group $L\_,$ and thus analogous to the other invariants from L-theory: the signature, a 4''k''-dimensional invariant (either symmetric or quadratic, $L^\; \backslash cong\; L\_$), and the De Rham invariant, a (4''k''+1)-dimensional ''symmetric'' invariant $L^.$ The Kervaire invariant problem is the problem of determining in which dimensions the Kervaire invariant can be nonzero. For differentiable manifolds, this can happen in dimensions 2, 6, 14, 30, 62, and possibly 126, and in no other dimensions. The final case of dimension 126 remains open. == Definition == The Kervaire invariant is the Arf invariant of the quadratic form determined by the framing on the middle-dimensional ''Z''/2''Z''-coefficient homology group :''q'' : ''H''2''m''+1(''M'';''Z''/2''Z'') $\backslash to$ ''Z''/2''Z'', and is thus sometimes called the Arf–Kervaire invariant. The quadratic form (properly, skew-quadratic form) is a quadratic refinement of the usual ε-symmetric form on the middle dimensional homology of an (unframed) even-dimensional manifold; the framing yields the quadratic refinement. The quadratic form ''q'' can be defined by algebraic topology using functional Steenrod squares, and geometrically via the self-intersections of immersions $S^$$\backslash to$ $M^$ determined by the framing, or by the triviality/non-triviality of the normal bundles of embeddings $S^$$\backslash to$ $M^$ (for $m\; \backslash neq\; 0,1,3$) and the mod 2 Hopf invariant of maps $S^\; \backslash to\; S^$ (for $m\; =\; 0,1,3$). 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「Kervaire invariant」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.