Postnikov tower

Summary

A Postnikov tower of a path-connected space $X$ is a sequence of spaces $X_1 \leftarrow X_2 \leftarrow X_3 \leftarrow \cdots$ with maps from $X$. The stage $X_n$ has the homotopy groups of $X$ up to degree $n$ and none above degree $n$. Each stage maps to the previous one by a fibration with fibre an Eilenberg–MacLane space. So the tower builds $X$ from Eilenberg–MacLane spaces, one homotopy group at a time. The extra data needed to rebuild $X$ are cohomology classes called $k$-invariants.