Postnikov system

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, and named after, Mikhail Postnikov.

The Postnikov system of a path-connected space "X" is a tower of spaces …→ "X_n" →…→ "X"₁→ "X"₀ with the following properties:

• each map "X_n"→"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 "X_n"→"X"_"n"−1, the fiber (call it "K"_"n") has a single homotopy group 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.

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