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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 Eilenberg–Mazur swindle ： ウィキペディア英語版 Eilenberg–Mazur swindle In mathematics, the Eilenberg–Mazur swindle, named after Samuel Eilenberg and Barry Mazur, is a method of proof that involves paradoxical properties of infinite sums. In geometric topology it was introduced by and is often called the Mazur swindle. In algebra it was introduced by Samuel Eilenberg and is known as the Eilenberg swindle or Eilenberg telescope (see telescoping sum). The Eilenberg–Mazur swindle is similar to the following well known joke "proof" that 1 = 0: : 1 = 1 + (−1 + 1) + (−1 + 1) + ... = 1 − 1 + 1 − 1 + ... = (1 − 1) + (1 − 1) + ... = 0 This "proof" is not valid as a claim about real numbers because Grandi's series 1 − 1 + 1 − 1 + ... does not converge, but the analogous argument can be used in some contexts where there is some sort of "addition" defined on some objects for which infinite sums do make sense, to show that if ''A'' + ''B'' = 0 then ''A'' = ''B'' = 0. ==Mazur swindle== In geometric topology the addition used in the swindle is usually the connected sum of knots or manifolds. Example : A typical application of the Mazur swindle in geometric topology is the proof that the sum of two non-trivial knots ''A'' and ''B'' is non-trivial. For knots it is possible to take infinite sums by making the knots smaller and smaller, so if ''A'' + ''B'' is trivial then :$A=A+(B+A)+(B+A)+\backslash cdots\; =\; (A+B)+(A+B)+\backslash cdots=0\backslash ,$ so ''A'' is trivial (and ''B'' by a similar argument). The infinite sum of knots is usually a wild knot, not a tame knot. See for more geometric examples. Example: The oriented ''n''-manifolds have an addition operation given by connected sum, with 0 the ''n''-sphere. If ''A'' + ''B'' is the ''n''-sphere, then ''A'' + ''B'' + ''A'' + ''B'' + ... is Euclidean space so the Mazur swindle shows that the connected sum of ''A'' and Euclidean space is Euclidean space, which shows that ''A'' is the 1-point compactification of Euclidean space and therefore ''A'' is homeomorphic to the ''n''-sphere. (This does not show in the case of smooth manifolds that ''A'' is diffeomorphic to the ''n''-sphere, and in some dimensions, such as 7, there are examples of exotic spheres ''A'' with inverses that are not diffeomorphic to the standard ''n''-sphere.) 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「Eilenberg–Mazur swindle」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.