In mathematics, the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in (Whitehead 1941).

The relevant MSC code is: 55Q15, Whitehead products and generalizations.

## Definition

Given elements $f\in \pi _{k}(X),g\in \pi _{l}(X)$ , the Whitehead bracket

$[f,g]\in \pi _{k+l-1}(X)\,$ is defined as follows:

The product $S^{k}\times S^{l}$ can be obtained by attaching a $(k+l)$ -cell to the wedge sum

$S^{k}\vee S^{l}$ ;

the attaching map is a map

$S^{k+l-1}{\stackrel {\phi }{\ \longrightarrow \ }}S^{k}\vee S^{l}.$ Represent $f$ and $g$ by maps

$f\colon S^{k}\to X$ and

$g\colon S^{l}\to X,$ then compose their wedge with the attaching map, as

$S^{k+l-1}{\stackrel {\phi }{\ \longrightarrow \ }}S^{k}\vee S^{l}{\stackrel {f\vee g}{\ \longrightarrow \ }}X.$ The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of

$\pi _{k+l-1}(X).$ Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so $\pi _{k}(X)$ has degree $(k-1)$ ; equivalently, $L_{k}=\pi _{k+1}(X)$ (setting L to be the graded quasi-Lie algebra). Thus $L_{0}=\pi _{1}(X)$ acts on each graded component.

## Properties

The Whitehead product is bilinear, graded-symmetric, and satisfies the graded Jacobi identity, and is thus a graded quasi-Lie algebra; this is proven in Uehara & Massey (1957) via the Massey triple product.

##### Relation to the action of $\pi _{1}$ If $f\in \pi _{1}(X)$ , then the Whitehead bracket is related to the usual action of $\pi _{1}$ on $\pi _{k}$ by

$[f,g]=g^{f}-g,$ where $g^{f}$ denotes the conjugation of $g$ by $f$ .

For $k=1$ , this reduces to

$[f,g]=fgf^{-1}g^{-1},$ which is the usual commutator in $\pi _{1}(X)$ . This can also be seen by observing that the $2$ -cell of the torus $S^{1}\times S^{1}$ is attached along the commutator in the $1$ -skeleton $S^{1}\vee S^{1}$ .

For a path connected H-space, all the Whitehead products on $\pi _{*}(X)$ vanish. By the previous subsection, this is a generalization of both the facts that the fundamental group of H-spaces are abelian, and that H-spaces are simple.

##### Suspension

All Whitehead products of classes $\alpha \in \pi _{i}(X)$ , $\beta \in \pi _{j}(X)$ lie in the kernel of the suspension homomorphism $\Sigma \colon \pi _{i+j-1}(X)\to \pi _{i+j}(\Sigma X)$ ## Examples

• $[\mathrm {id} _{S^{2}},\mathrm {id} _{S^{2}}]=2\cdot \eta \in \pi _{3}(S^{2})$ , where $\eta \colon S^{3}\to S^{2}$ is the Hopf map.

This can be shown by observing that the Hopf invariant defines an isomorphism $\pi _{3}(S^{2})\cong \mathbb {Z}$ and explicitly calculating the cohomology ring of the cofibre of a map representing $[\mathrm {id} _{S^{2}},\mathrm {id} _{S^{2}}]$ .