Postnikov's algebraic reconstruction of cohomology from homotopy invariants

Question

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).

Answer 1

First, you say:

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.

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 $G$.

Answer 2

One question which has puzzled me is, just to take the $2$-type, how does one specify an element of $H^3(\pi_1,\pi_2)$?

In 1972, Philip Higgins and I discussed with Saunders Mac Lane the possibility of a $2$-dimensional van Kampen theorem to calculate the homotopy $2$-type of a union, and he explained why he had decided this was impossible. Calculating $\pi_1$ is OK, by the van Kampen theorem. Then you have to calculate $\pi_2$, as a $\pi_1$-module; then the $k$-invariant! Absurd!

Now the $2$-type is also described by a crossed module, by a theorem of Mac Lane and Whitehead. Philip and I proved in 1975, in a paper published in 1978, that under reasonable circumstances, the crossed module of a union is given by a pushout of crossed modules. That seems some kind of answer!

Of course, there is still a problem of calculating $\pi_2$ from this pushout, and even more of calculating the $k$-invariant. But at least the topology has been translated into algebra; the theorem generalises strongly a theorem of Whitehead on free crossed modules, and has enabled specific, even computer, calculations of $2$-types. For more detail, see this 2011 EMS Tract vol 15.

In effect, crossed modules are a useful algebraic model of homotopy $2$-types, and it is certainly useful to consider what one wants from an algebraic model.

Thus Loday worried that his cat$^n$-group model (called by him $n$-cat groups in his 1982 paper) of homotopy $(n+1)$-types was "purely formal", so was very happy with our van Kampen theorem for $n$-cubes of spaces which enabled some new computations, for example of homotopy $3$-types of a suspension of a $K(G,1)$. More applications and related algebra were developed by Ellis and Steiner, in a 1987 JPAA paper.

It also seems true that strict $n$-fold groupoids model weak homotopy $n$-types. How does one use this? and how is it related to Postnikov systems?

To add a bit to a possible answer to the question on cohomology, Graham Ellis has a paper on the cohomology of crossed modules.

Let the debate continue!

May 7, 2019 Looking at my answer, I first updated two links and then realised there is more to be said on modelling homotopy types, part of which is in this paper, published in 2018.

The general problem of relating cohomology to models of homotopy types seems very difficult.

May 23, 2019 I should also have referred you to Loday's paper ‘Spaces with finitely many nontrivial homotopy groups’. J. Pure Appl. Algebra 24 (2) (1982) 179–202. and Ellis, G. J. and Steiner, R. ‘Higher-dimensional crossed modules and the homotopy groups of (n + 1)-ads’. J. Pure Appl. Algebra 46 (2-3) (1987) 117–136.