---
tag: D3649AJ30F
kind: concept
name: Whitehead tower
msc: [55S45, 55P20]
sources:
  - url: https://en.wikipedia.org/wiki/Postnikov_system
---

## Summary

The Whitehead tower of a space $X$ is a sequence of spaces $\cdots \to X\langle 2 \rangle \to X\langle 1 \rangle \to X$.
The stage $X\langle n \rangle$ is $n$-connected and has the same homotopy groups as $X$ above degree $n$.
It is dual to the [[Postnikov tower|EKGEZGHYBW]]: instead of discarding the high homotopy groups, it discards the low ones.
Its first stage is the [[universal cover|Y1YPE6PM52]], and each further stage is a fibration over the previous one with fibre an [[Eilenberg–MacLane space|AJHEYJE5XY]].
It is used to compute homotopy groups with spectral sequences and to define string structures through the stage $BO\langle 8 \rangle$.

## Explanations

### L1 human

The Whitehead tower peels off the homotopy groups of a space from the bottom.
The first step replaces the space by its universal cover, which removes the fundamental group.
The next step removes the second homotopy group, and so on.
Each stage keeps all the higher homotopy groups unchanged.
For example, the $3$-connected cover of the $3$-sphere has the same homotopy groups as $S^3$ in degrees $4$ and above, but none below.