# Whitehead product

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
    $\pi _{k+l-1}(X).$

## Grading

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 satisfies the following properties:
* Bilinearity. $[f,g+h]=[f,g]+[f,h],[f+g,h]=[f,h]+[g,h]$
* Graded Symmetry. $[f,g]=(-1)^{pq}[g,f],f\in \pi _{p}X,g\in \pi _{q}X,p,q\geq 2$
* Graded Jacobi identity. $(-1)^{pr}[[f,g],h]+(-1)^{pq}[[g,h],f]+(-1)^{rq}[[h,f],g]=0,f\in \pi _{p}X,g\in \pi _{q}X,h\in \pi _{r}X{\text{ with }}p,q,r\geq 2$
Sometimes the homotopy groups of a space, together with the Whitehead product operation are called 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 $S^{1}\vee S^{1}$.

### Whitehead products on H-spaces

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 groups 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 $[\mathrm {id} _{S^{2}},\mathrm {id} _{S^{2}}]$. Using the Pontryagin–Thom construction there is a direct geometric argument, using the fact that the preimage of a regular point is a copy of the Hopf link.