Postnikov tower ($\{P_{n}X\}$)

A Postnikov tower of a connected CW complex $X$ consists of spaces $P_{n}X$ with ${\pi }_i(P_{n}X)={\pi }_i(X)$ for $i\le n$ and zero for $i>n$, together with compatible maps $X\to P_{n}X\to P_{n-1}X$. For simply connected $X$, each stage is a principal Eilenberg–MacLane fibration with fibre $K({\pi }_{n}X,n)$, classified by a Postnikov invariant. The canonical map $X\to {\mathrm{holim}}_n{P}_{n}X$ is a weak equivalence: in each homotopy degree the tower is eventually constant, so the derived inverse-limit obstruction vanishes.

For an abelian group $A$, a principal fibration with fibre $K(A,n)$ is a homotopy pullback of the path fibration $PK(A,n+1)\to K(A,n+1)$. It is therefore the homotopy fibre of a map $k:B\to K(A,n+1)$ and is classified by $[k]\in H^{n+1}(B;A)$. The fibre acts by translation up to homotopy; in this untwisted description the fundamental group acts trivially on $A$. This is the principal-fibration form used for simply connected Postnikov towers.

The $n$th Postnikov invariant is the cohomology class $k_n\in H^{n+1}(P_{n-1}X;{\pi }_{n}X)$ whose representing map has homotopy fibre $P_{n}X$. It specifies how the new homotopy group is attached to the previous stage. A zero invariant gives the product $P_{n}X\simeq P_{n-1}X\times K({\pi }_{n}X,n)$. Merely tensoring homotopy groups with $\mathbb{Q}$ does not describe rationalization of a topological space unless these classes are also transported.