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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 Ganea conjecture ： ウィキペディア英語版 Ganea conjecture Ganea's conjecture is a claim in algebraic topology, now disproved. It states that : $\backslash text(X\; \backslash times\; S^n)=\backslash text(X)\; +1,\; n>0\; \backslash ,\backslash !$ where cat(''X'') is the Lusternik–Schnirelmann category of a topological space ''X'', and ''S''''n'' is the ''n'' dimensional sphere. The inequality : $\backslash text(X\; \backslash times\; Y)\; \backslash le\; \backslash text(X)\; +\backslash text(Y)$ holds for any pair of spaces, ''X'' and ''Y''. Furthermore, cat(''S''''n'')=1, for any sphere ''S''''n'', ''n''>0. Thus, the conjecture amounts to cat(''X'' × S''n'') ≥ cat(''X'') + 1. The conjecture was formulated by Tudor Ganea in 1971. Many particular cases of this conjecture were proved, till finally Norio Iwase gave a counterexample in 1998. In a follow-up paper from 2002, Iwase gave an even stronger counterexample, with ''X'' a closed, smooth manifold. This counterexample also disproved a related conjecture, stating that : $\backslash text(M\; -\; )=\backslash text(M)\; -1\; ,$ for a closed manifold ''M'' and ''p'' a point in ''M''. This work raises the question: For which spaces ''X'' is the Ganea condition, cat(''X'' × S''n'') = cat(''X'') + 1, satisfied? It has been conjectured that these are precisely the spaces ''X'' for which cat(''X'') equals a related invariant, Qcat(''X''). ==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「Ganea 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.