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 by, and named after, Mikhail Postnikov.

The Postnikov system of a path-connected space X is a tower of spaces …→ X_n →…→ X₁→ X₀ 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 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

• Postnikov, Mikhail M. (1951). "[Determination of the homology groups of a space by means of the homotopy invariants]". Doklady Akademii Nauk SSSR 76: 359–362.
• Hatcher, Allen (2002). Algebraic topology. Cambridge University Press. ISBN 978-0-521-79540-1.