Postnikov system

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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 →…→ X1X0 with the following properties:

  • each map XnXn−1 is a fibration;
  • πk(Xn) = πk(X) for kn;
  • πk(Xn) = 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 = limn Xn. By the long exact sequence for the fibration XnXn−1, the fiber (call it Kn) has at most one non-trivial homotopy group, which will be in degree n; it is thus an Eilenberg–MacLane space of type Kn(X), n). The Postnikov system can be thought of as a way of constructing X out of Eilenberg–MacLane spaces.