In his short paper (1951) and longer monograph (1955), Postnikov introduced what I believe are now called Postnikov systems or towers. It is my understanding that Postnikov systems have since then been widely adopted, as a way of totally encoding the homotopy type of a topological space. Expressed algebraically, a Postnikov system of a topological space $X$ consists of its homotopy groups $\pi_i = \pi_i(X)$, where each of the higher groups $\pi_i$ ($i\ge 2$) has the structure of a $\pi_1$-module, together with a group cohomology class $[k_1] \in H^{3}(\pi_1,\pi_2)$ and higher "cohomology classes" (that are somewhat more difficult to describe algebraically) $[k_i]$ with coefficients in $\pi_{i+1}$ for each $i\ge 2$.
The main application of these systems, in Postnikov's original work, was to reconstruct the cohomology of the space $X$ from its homotopy invariants in a purely algebraic way. The reconstruction allowed also for cohomology with coefficients in a local system, with the local system algebraically presented as some $\pi_1$-module. In that work, this result was presented as a generalization of the earlier work (1945) of Eilenberg and MacLane (and of course others) on the algebraic reconstruction of the cohomology of spaces that are aspherical except in one degree.
Now my question. Is there a modern reference for the reconstruction of the cohomology (desirably with local coefficients) of a topological space from its Postnikov sequence? While Postnikov sequences themselves are treated in many places, I've not been able to find the cohomology reconstruction theorem anywhere except in Postnikov's original longer monograph (in Russian).
Best Answer
First, you say:
I don't think that this is really true. There are very few cases where one can actually describe the full Postnikov system explicitly, so it is not widely used, except for certain very restricted classes of spaces.
Next, in thinking about what is theoretically possible, you should remember the Kan-Thurston theorem. That says that for any connected space $X$ there is a group $G$ and a map $K(G,1)\to X$ that induces an isomorphism in homology. Here the space $K(G,1)$ has the simplest possible Postnikov tower, with only one layer and no $k$-invariants, but the homology can be anything you want, and does not depend in any very obvious way on $G$.