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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 Novikov conjecture ： ウィキペディア英語版 Novikov conjecture :''This page concerns mathematician Sergei Novikov's topology conjecture. For astrophysicist Igor Novikov's conjecture regarding time travel, see Novikov self-consistency principle. The Novikov conjecture is one of the most important unsolved problems in topology. It is named for Sergei Novikov who originally posed the conjecture in 1965. The Novikov conjecture concerns the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold, arising from the fundamental group. According to the Novikov conjecture, the ''higher signatures'', which are certain numerical invariants of smooth manifolds, are homotopy invariants. The conjecture has been proved for finitely generated abelian groups. It is not yet known whether the Novikov conjecture holds true for all groups. There are no known counterexamples to the conjecture. ==Precise formulation of the conjecture== Let ''G'' be a discrete group and ''BG'' its classifying space, which is a K(G,1) and therefore unique up to homotopy equivalence as a CW complex. Let :$f:\; M\backslash rightarrow\; BG$ be a continuous map from a closed oriented ''n''-dimensional manifold ''M'' to ''BG'', and :$x\; \backslash in\; H^\; (BG;\backslash mathbb\; ).$ Novikov considered the numerical expression, found by evaluating the cohomology class in top dimension against the fundamental class (), and known as a higher signature: :$\backslash left\backslash langle\; f^$ *(x) \cup L_i(M),() \right\rangle \in \mathbb where ''L''i is the ''i''th Hirzebruch polynomial, or sometimes (less descriptively) as the ''i''th ''L''-polynomial. For each ''i'', this polynomial can be expressed in the Pontryagin classes of the manifold's tangent bundle. The Novikov conjecture states that the higher signature is an invariant of the oriented homotopy type of ''M'' for every such map ''f'' and every such class ''x'', in other words, if $h:\; M\text{'}\; \backslash rightarrow\; M$ is an orientation preserving homotopy equivalence, the higher signature associated to $f\; \backslash circ\; h$ is equal to that associated to ''f''. 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「Novikov conjecture」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.