Postnikov system

In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space by filtering its homotopy type. What this looks like is for a space ${\displaystyle X}$ there is a list of spaces ${\displaystyle \{X_{n}\}_{n\geq 0}}$ where

${\displaystyle \pi _{k}(X_{n})={\begin{cases}\pi _{k}(X)&{\text{ for }}k\leq n\\0&{\text{ for }}k>n\end{cases}}}$

and there's a series of maps ${\displaystyle \phi _{n}:X_{n}\to X_{n-1}}$ that are fibrations with fibers Eilenberg-MacLane spaces ${\displaystyle K(\pi _{n}(X),n)}$. In short, we are decomposing the homotopy type of ${\displaystyle X}$ using an inverse system of topological spaces whose homotopy type at degree ${\displaystyle k}$ agrees with the truncated homotopy type of the original space ${\displaystyle X}$. Postnikov systems were introduced by, and are named after, Mikhail Postnikov.

There is a similar construction called the Whitehead tower (defined below) where instead of having spaces ${\displaystyle X_{n}}$ with the homotopy type of ${\displaystyle X}$ for degrees ${\displaystyle \leq n}$, these spaces have null homotopy groups ${\displaystyle \pi _{k}(X_{n})=0}$ for ${\displaystyle 1<k<n}$.