# $k$-invariants as extension classes

Let $X$ be a connected cell complex with fundamental group $G$ and $(n-1)$-connected universal covering space. Let $\Pi=\pi_n(X)$. We may construct a $K(G,1)$ complex $K$ by adjoining cells of dimension $>n$ to $X$. Let $c_X:X\to{K}$ be the inclusion. If we view $c_X$ as a fibration, the $k$-invariant $k_1(X)$ is the first obstruction to the existence of a section for $c_X$.

Question: Can $k_1(X)$ be identified with the extension class of the iterated extension $0\to\Pi\to{C_n/dC_{n+1}}\to\dots\to{C_0}\to\mathbb{Z}\to0$, as elements of $H^{n+1}(G;\Pi)$?

This may well be somewhere in the papers of Eilenberg, Mac Lane and/or JHC Whitehead from the late 1940s, but I had not been able to track down a published proof. (The Homotopy Addition Theorem may be used to identify the homological extension class with the first obstruction to retracting $K$ onto $X$, but that is not quite the same problem.)

This is correct, and can be seen as follows: The main idea is to model the map from the homotopy groups of the universal cover of $X$ to its homology as a map of spaces of $G$-action, and then use naturality of the k-classes with respect to morphisms.
We have a morphism of spaces $P_{\le n} X\to P_{\le n} \mathbb{Z} \otimes X$, were here $P_{\le n}$ is the $n$-th Postnikov piece and $\otimes \mathbb{Z}$ replaces a space by the space associated via Dold Kan correspondence with its chain complex. Since the Dold Kan correspondence is a fully faithful embedding of chain complexes in spaces mapping $G[n]$ to $K(G,n)$, the $K$-classes are computed the same for a chain complex in chain complexes (which is the extension you suggested) and in spaces (which is the usual K-class).

In order to get what you want, because you have non-trivial foundamental group, what we need though is an equivariant version. Namely, we have the same picture with spaces replaced by $G$-equivariant spaces and complexes by complexes of $G$-modules. Then, using the fact that the morphism from the universal cover to the Dold Kan correspondent induces an isomorphism on $\pi_n$ we can reduce the problem to a $G$-equivariant space which is in the image of the Dold Kan correspondence and then compute in complexes.

I think what you are looking for at least in terms of mappings rather than sections is Theorem 12.2.10 of the book Nonabelian AlgebraicTopology (NAT), which is phrased in terms of the homotopy theory of crossed complexes. Maybe this needs to be developed in terms of sections.

(Obstruction Class Theorem) Let $n \geqslant 2$ and let $F,C$ be reduced crossed complexes such that $F$ is free, $C$ is $n$-aspherical, and $C_i=0$ for $i >n$. Let $G= \pi_1 F, H= \pi_1 C, M=$ Ker $\delta_n \colon C_n \to C_{n-1}$. Let $\theta \colon G \to H$ be a morphism of groups. Then there is defined an element $k_\theta \in H^{n+1}_{\theta\phi}(F,M)$, called the obstruction class of $\theta$, such that the vanishing of $k_\theta$ is necessary and sufficient for $\theta$ to be realised by a morphism $F \to C$.

If $k_\theta=0$, then the set $[F,C;\theta\phi]$ of homotopy classes of morphisms $F\to C$ realising $\theta \phi$ is bijective with $H^n_{\theta\phi}(F,M)$. $\square$

A reduced crossed complex $C$ is one which $C_0$ is a singleton.

The NAT book also spells out the adjoint relation of crossed complexes and chain complexes with a groupoid of operators. All this is related to the answer of S. carmeli.

The other author you could look at is Johannes Huebschmann, who developed work on crossed complexes and group cohomology, which also has relations to earlier work of A.S.-T. Lue on cohomomology with respect to a variety, see references [Hue...], and [Lue81], in NAT.