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. For a space X {\displaystyle X} , this is a list of spaces { X n } n ≥ 0 {\displaystyle \{X_{n}\}_{n\geq 0}} where π k ( X n ) = { π k ( X ) for k ≤ n 0 for k > n {\displaystyle \pi _{k}(X_{n})={\begin{cases}\pi _{k}(X)&{\text{ for }}k\leq n\\0&{\text{ for }}k>n\end{cases}}} and a series of maps ϕ n : X n → X n − 1 {\displaystyle \phi _{n}:X_{n}\to X_{n-1}} that are fibrations with Eilenberg-MacLane spaces K ( π n ( X ) , n ) {\displaystyle K(\pi _{n}(X),n)} as fibers. In short, we are decomposing the homotopy type of X {\displaystyle X} using an inverse system of topological spaces whose homotopy type at degree k {\displaystyle k} agrees with the truncated homotopy type of the original space X {\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 X n {\displaystyle X_{n}} with the homotopy type of X {\displaystyle X} for degrees ≤ n {\displaystyle \leq n} , these spaces have null homotopy groups π k ( X n ) = 0 {\displaystyle \pi _{k}(X_{n})=0} for 1 < k < n {\displaystyle 1<k<n} .
Definition A Postnikov system of a path-connected space X {\displaystyle X} is an inverse system of spaces
⋯ → X n → p n X n − 1 → p n − 1 ⋯ → p 3 X 2 → p 2 X 1 → p 1 ∗ {\displaystyle \cdots \to X_{n}\xrightarrow {p_{n}} X_{n-1}\xrightarrow {p_{n-1}} \cdots \xrightarrow {p_{3}} X_{2}\xrightarrow {p_{2}} X_{1}\xrightarrow {p_{1}} *}
with a sequence of maps ϕ n : X → X n {\displaystyle \phi _{n}:X\to X_{n}} compatible with the inverse system such that
The map ϕ n : X → X n {\displaystyle \phi _{n}:X\to X_{n}} induces an isomorphism π i ( X ) → π i ( X n ) {\displaystyle \pi _{i}(X)\to \pi _{i}(X_{n})} for every i ≤ n {\displaystyle i\leq n} .
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