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 …→ Xn →…→ X1→ X0 with the following properties:
- each map Xn→Xn−1 is a fibration;
- πk(Xn) = πk(X) for k ≤ n;
- 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 Xn→Xn−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 K(πn(X), n). The Postnikov system can be thought of as a way of constructing X out of Eilenberg–MacLane 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&ndash, 362.
- Hatcher, Allen (2002). Algebraic topology. Cambridge University Press. ISBN 978-0-521-79540-1.