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, 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 a single homotopy group in degree n; it is thus an Eilenberg-Mac Lane space of type K(πn(X), n). The Postnikov system can be thought of as a way of constructing X out of Eilenberg-Mac Lane spaces.
[edit] References
- Postnikov, Mikhail (1951), "Determination of the homology groups of a space by means of the homotopy invariants", Doklady Akad. Nauk SSSR (N.S.) 76: 359–362
- Hatcher, Allen (2002), Algebraic topology, Cambridge University Press, ISBN 978-0-521-79540-1, http://www.math.cornell.edu/~hatcher/AT/ATpage.html
