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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 Hauptvermutung ： ウィキペディア英語版 Hauptvermutung The Hauptvermutung (German for main conjecture) of geometric topology is the conjecture that any two triangulations of a triangulable space have a common refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by Steinitz and Tietze. This conjecture is now known to be false. The non-manifold version was disproved by John Milnor in 1961 using Reidemeister torsion. The manifold version is true in dimensions . The cases were proved by Tibor Radó and Edwin E. Moise in the 1920s and 1950s, respectively. An obstruction to the manifold version was formulated by Andrew Casson and Dennis Sullivan in 1967–9 (originally in the simply-connected case), using the Rochlin invariant and the cohomology group H3(''M'';Z/2Z). A homeomorphism of ''m''-dimensional piecewise linear manifolds has an invariant such that for , ƒ is isotopic to a piecewise linear (PL) homeomorphism if and only if . In the simply-connected case and with , ƒ is homotopic to a PL homeomorphism if and only if The obstruction to the manifold Hauptvermutung is now seen as a relative version of the triangulation obstruction of Rob Kirby and Larry Siebenmann, obtained in 1970. The Kirby–Siebenmann obstruction is defined for any compact ''m''-dimensional topological manifold ''M'' :$\backslash kappa(M)\backslash in\; H^4(M;\backslash mathbb/2\backslash mathbb)$ again using the Rochlin invariant. For , ''M'' has a PL structure (i.e. can be triangulated by a PL manifold) if and only if , and if this obstruction is 0 the PL structures are parametrized by H3(''M'';Z/2Z). In particular there are only a finite number of essentially distinct PL structures on ''M''. For compact simply-connected manifolds of dimension 4, Simon Donaldson found examples with an infinite number of inequivalent PL structures, and Michael Freedman found the E8 manifold which not only has no PL structure, but (by work of Casson) is not even homeomorphic to a simplicial complex. In 2013, Ciprian Manolescu proved that there exist compact manifolds of dimension 5 (and hence of any dimension greater than 5) that are not homeomorphic to a simplicial complex. == References == 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「Hauptvermutung」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.