---
tag: WP12H9FG91
kind: definition
name: Whitehead product
msc: [55Q15]
defines: QKEE8AT2NR
uses: [P976WYEFMB, 5JSWM1HBX2, 5W95QFHXPR, T9DG8DDE75]
sources:
  - url: https://en.wikipedia.org/wiki/Whitehead_product
---

## Definition

Let $(X, x_0)$ be a [[pointed space|T9DG8DDE75]], and let $p, q \ge 1$.
Give $S^p \times S^q$ the [[CW structure|5W95QFHXPR]] with cells of dimensions $0$, $p$, $q$ and $p + q$, so that its $(p+q-1)$-skeleton is the [[wedge sum|5JSWM1HBX2]] $S^p \vee S^q$.
Let $w \colon S^{p+q-1} \to S^p \vee S^q$ be the attaching map of the top cell.
For $\alpha = [f] \in \pi_p(X, x_0)$ and $\beta = [g] \in \pi_q(X, x_0)$, the *Whitehead product* is
$$[\alpha, \beta] = [(f \vee g) \circ w] \in \pi_{p+q-1}(X, x_0),$$
where $f \vee g \colon S^p \vee S^q \to X$ is $f$ on $S^p$ and $g$ on $S^q$, and $\pi_k$ are the [[homotopy groups|P976WYEFMB]].

## Remarks

The class $[\alpha, \beta]$ depends only on $\alpha$ and $\beta$.
For $p = q = 1$, the map $w$ traces the boundary of the square in the torus, and $[\alpha, \beta]$ is the commutator $\alpha \beta \alpha^{-1} \beta^{-1}$, up to the choice of conventions.

Example: let $\iota_2 \in \pi_2(S^2)$ be the class of the identity.
Then $[\iota_2, \iota_2] \in \pi_3(S^2) \cong \Z$ has Hopf invariant $\pm 2$, so it is twice a generator, up to sign.

Non-example of a nonzero product: if $X$ is an H-space, for instance a topological group, then all Whitehead products in $X$ vanish.
In particular, every Whitehead product in $S^1$ or $S^3$ is zero.

For $p, q \ge 2$ the product is bilinear and satisfies $[\alpha, \beta] = (-1)^{pq} [\beta, \alpha]$, together with a graded Jacobi identity.