nLab
Postnikov system

Definition

A Postnikov system or Postnikov tower (M M Postnikov, 1951) is a sequence of path connected, pointed topological spaces $X^{(n)}$, $n\ge 1$, such that ${\pi }_r(X^{(n)})=0$ for $r>n$, together with a sequence ${\pi }_n$ of modules of the fundamental group ${\pi }_1(X^{(1)})$ of $X^{(1)}$ and fibrations $p_n:X^{(n+1)}\to X^{(n)}$ classified up to homotopy type by a specified cohomology class $k^{n+1}\in H^{n+1}(X^{(n)},{\pi }_n)$.

Properties

It is known that Postnikov systems classify all weak, pointed connected homotopy types. In particular, if $X$ satisfies ${\pi }_r(X)=0$ for $1<r<n$ then the first non trivial Postnikov invariant is an element $k^{n+1}$ of group cohomology (with twisted coefficients of course). Such elements are also determined by $n$-fold crossed extensions of ${\pi }_n$ by ${\pi }_1$, which are exact crossed complexes of the form

$$0\to {\pi }_n\to C_n\to C_{n-1}\to \cdots \to C_2\to C_1$$0 \to \pi_n \to C_n \to C_{n-1} \to \cdots \to C_2 \to C_1

together with an isomorphism $\mathrm{Coker}(C_2\to C_1)\cong {\pi }_1$. This gives an algebraic model of such an $n$-type. Advantages of algebraic models are that algebraic constructions can be made on them, such as forming limits or colimits. The various higher homotopy van Kampen theorems are useful in the latter case. For example, it may be difficult or well nigh impossible to write down a determination of the Postnikov invariant of a pushout of crossed modules, even if the pushout consists of finite groups.

A Postnikov system is easiest to understand in the 2-stage case, i.e. two non vanishing homotopy groups, and focuses attention on the cohomology of Eilenberg-Mac Lane spaces, which also determine all cohomology operations. Basic work on this area was done by Eilenberg and Mac Lane, and by H. Cartan, while the theory of cohomology operations, including Steenrod operations, is itself a large area.

The reference below shows the problems in the 3-stage systems.

For homotopy 3-types, the algebraic model of crossed squares is more explicit than the corresponding Postnikov system, and more calculable. However, not much work has been done on, say, cohomology operations using the algebraic model of $n$-fold groupoids, and it is not clear if that would help.

Generalizations

There are analogues in other setups, e.g.

• Postnikov systems in triangulated categories

• motivic homotopy theory (M. Levine, The Postnikov tower in motivic stable homotopy theory).

• Postnikov tower in an (∞,1)-category

References

• M.M. Postnikov, Determination of the homology groups of a space by means of the homotopy invariants, Doklady Akad. Nauk SSSR (N.S.) 76: 359–362 (1951)
• George Whitehead, Elements of homotopy theory, chapter 9
• Donald W. Kahn, The spectral sequence of a Postnikov system, Comm. Math. Helv. 40, n.1, 169–198, 1965 doi
• P. I. Booth, An explicit classification of three-stage Postnikov towers, Homology, homotopy and applications 8 (2006), No. 2, 133–155
• G. Ellis and R. Mikhailov, A colimit of classifying spaces, arXiv:0804.3581.

A pedagogical introduction to Postnikov systems with an eye towards their $\mathrm{\infty }$-groupoid incarnation under the correspondence given by the homotopy hypothesis is in

• John Baez, Mike Shulman, Lectures on n-Categories and Cohomology