Whitehead product

Builds on Homotopy group Wedge sum CW complex Pointed space

Definition

Let $(X, x_0)$ be a pointed space, and let $p, q \ge 1$ . Give $S^p \times S^q$ the CW structure with cells of dimensions $0$ , $p$ , $q$ and $p + q$ , so that its $(p+q-1)$ -skeleton is the wedge sum $S^p \vee S^q$ . 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.

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.

Reading path

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CW complex concept
Homotopy group concept
Pointed space concept
Wedge sum concept
Whitehead product definition

Tag	`WP12H9FG91`
URI	`m2://WP12H9FG91`
Kind	definition
Writing	W0 stub
MSC	55Q15
Defines	Whitehead product
Uses	Homotopy group, Wedge sum, CW complex, Pointed space
Changed	2 November 2026

Whitehead product

Definition

Remarks

Reading path

Further reading

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