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In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of constructing a topological space from its homotopy groups. Postnikov systems were introduced by, and named after, Mikhail Postnikov. The Postnikov system of a path-connected space ''X'' is a tower of spaces …→ ''Xn'' →…→ ''X''1→ ''X''0 with the following properties: * each map ''Xn''→''X''''n''−1 is a fibration; * π''k''(''X''''n'') = π''k''(''X'') for ''k'' ≤ ''n''; * π''k''(''X''''n'') = 0 for ''k'' > ''n''. Every path-connected space has such a Postnikov system, and it is unique up to homotopy. The space ''X'' can be reconstructed from the Postnikov system as its inverse limit: ''X'' = lim''n'' ''X''''n''. By the long exact sequence for the fibration ''Xn''→''X''''n''−1, the fiber (call it ''K''''n'') has at most one non-trivial homotopy group, which will be in degree ''n''; it is thus an Eilenberg–Mac Lane space of type ''K''(π''n''(''X''), ''n''). The Postnikov system can be thought of as a way of constructing ''X'' out of Eilenberg–Mac Lane spaces. ==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「Postnikov system」の詳細全文を読む スポンサード リンク
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