---
tag: QKEE8AT2NR
kind: concept
name: Whitehead product
msc: [55Q15]
sources:
  - url: https://en.wikipedia.org/wiki/Whitehead_product
same_as:
  wikidata: Q7996042
  wikipedia: Whitehead product
---

## Summary

The Whitehead product is an operation $[\alpha, \beta] \in \pi_{p+q-1}(X)$ on [[homotopy groups|P976WYEFMB]], defined for $\alpha \in \pi_p(X)$ and $\beta \in \pi_q(X)$.
It comes from the attaching map of the top cell of the product $S^p \times S^q$.
For $p = q = 1$ it is the commutator in the fundamental group.
With it, the homotopy groups of a space form a graded Lie algebra, up to a shift of degree and signs.

## Explanations

### L1 human

The product of two spheres $S^p \times S^q$ is built from the wedge $S^p \vee S^q$ by gluing in one more cell.
The gluing map is a sphere of dimension $p + q - 1$ sitting in the wedge.
Given two spheres in a space $X$, sending that gluing map into $X$ gives a new element of a higher homotopy group: their Whitehead product.
For two loops on a torus the product is trivial, which reflects that the two loops commute there.