Postnikov tower

Builds on Path-connected space Homotopy group Eilenberg–MacLane space

Definition

Let $X$ be a path-connected space with base point, and let $\pi_k$ denote the $k$ -th homotopy group. A Postnikov tower of $X$ is a sequence of pointed path-connected spaces $X_n$ , $n \ge 1$ , with pointed maps $f_n \colon X \to X_n$ and $p_{n} \colon X_{n} \to X_{n-1}$ for $n \ge 2$ , such that:

$p_n \circ f_n = f_{n-1}$ for all $n \ge 2$ .
$f_n$ induces an isomorphism $\pi_k(X) \to \pi_k(X_n)$ for all $k \le n$ .
$\pi_k(X_n) = 0$ for all $k > n$ .

Remarks

Every path-connected CW complex has a Postnikov tower, built by attaching cells of dimension at least $n + 2$ to $X$ to kill $\pi_k$ for $k > n$ . Up to homotopy, $p_n$ can be taken to be a fibration, and then its fibre is an Eilenberg–MacLane space $K(\pi_n(X), n)$ . The map from $X$ to the homotopy limit of the tower is a weak homotopy equivalence.

Example: let $X = K(G, m)$ . Take $X_n$ a point for $n < m$ and $X_n = X$ for $n \ge m$ , with the evident maps. Clause 2 holds since $\pi_k(X) = 0$ for $k < m$ , and clause 3 holds since $\pi_k(X) = 0$ for $k > m$ .

Non-example: for $X = S^2$ , the choice $X_2 = S^2$ with $f_2 = \id$ fails clause 3, since $\pi_3(S^2) \cong \Z \ne 0$ . A valid choice is $X_2 = \C P^\infty$ , a $K(\Z, 2)$ .

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Tag	`KC7BVZZZ2W`
URI	`m2://KC7BVZZZ2W`
Kind	definition
Writing	W0 stub
MSC	55S45, 55P20
Defines	Postnikov tower
Uses	Path-connected space, Homotopy group, Eilenberg–MacLane space
Changed	2 November 2026

Postnikov tower

Definition

Remarks

Reading path

Further reading

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