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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 Wall's finiteness obstruction ： ウィキペディア英語版 Wall's finiteness obstruction In geometric topology, a field within mathematics, the obstruction to a finitely dominated space ''X'' being homotopy-equivalent to a finite CW-complex is its Wall finiteness obstruction ''w(X)'' which is an element in the reduced zeroth algebraic K-theory of the integral group ring $\backslash widetilde\_0(\backslash mathbb())$. It is named after the mathematician C. T. C. Wall. By Milnor's work〔Milnor, J. ''On spaces having the homotopy type of a CW-complex''. Transactions of the American Mathematical Society Vol. 90, No. 2 (Feb., 1959), pp. 272-280.〕 on finitely dominated spaces, no generality is lost in letting ''X'' be a CW-complex. A ''finite domination'' of ''X'' is a finite CW-complex ''K'' together with maps $r:K\backslash to\; X$ and $i:X\backslash to\; K$ such that $r\backslash circ\; i\backslash simeq\; 1\_X$. By a construction due to Milnor it is possible to extend ''r'' to a homotopy equivalence $\backslash bar:\backslash bar\backslash to\; X$ where $\backslash bar$ is a complex obtained from ''K'' by attaching cells to kill the relative homotopy groups $\backslash pi\_n(r)$. $\backslash bar$ will be ''finite'' if all relative homotopy groups are finitely generated. Wall showed that this will be the case if and only if his finiteness obstruction vanishes. More precisely, using covering space theory and the Hurewicz theorem one can identify $\backslash pi\_n(r)$ with $H\_n(\backslash widetilde,\backslash widetilde)$. Wall then showed that the cellular chain complex $C\_$ *(\widetilde) is chain-homotopy equivalent to a chain complex $A\_$ * of finite type of projective $\backslash mathbb()$-modules, and that $H\_n(\backslash widetilde,\backslash widetilde)\backslash cong\; H\_n(A\_$ *) will be finitely generated if and only if these modules are stably-free. Stably-free modules vanish in reduced K-theory. This motivates the definition :$w(X)=\backslash sum\_i(-1)^i()\backslash in\backslash widetilde\_0(\backslash mathbb())$. ==See also== *Algebraic K-theory *Whitehead torsion 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「Wall's finiteness obstruction」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.