Whitehead product

Grading

Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so ${\displaystyle {\pi }_k(X)}$ has degree ${\displaystyle (k-1)}$; equivalently, ${\displaystyle L_k={\pi }_{k+1}(X)}$ (setting L to be the graded quasi-Lie algebra). Thus ${\displaystyle L_0={\pi }_1(X)}$ acts on each graded component.

Properties

The Whitehead product satisfies the following properties:

• Bilinearity. ${\displaystyle [f,g+h]=[f,g]+[f,h],[f+g,h]=[f,h]+[g,h]}$
• Graded Symmetry. ${\displaystyle [f,g]=(-1{)}^{pq}[g,f],f\in {\pi }_{p}X,g\in {\pi }_{q}X,p,q\ge 2}$
• Graded Jacobi identity. ${\displaystyle (-1{)}^{pr}[[f,g],h]+(-1{)}^{pq}[[g,h],f]+(-1{)}^{rq}[[h,f],g]=0,f\in {\pi }_{p}X,g\in {\pi }_{q}X,h\in {\pi }_{r}X\text{ with }p,q,r\ge 2}$

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 π 1 {\displaystyle \pi _{1}}

If ${\displaystyle f\in {\pi }_1(X)}$, then the Whitehead bracket is related to the usual action of ${\displaystyle {\pi }_1}$ on ${\displaystyle {\pi }_k}$ by

${\displaystyle [f,g]=g^f-g,}$

where ${\displaystyle g^f}$ denotes the conjugation of ${\displaystyle g}$ by ${\displaystyle f}$.

For ${\displaystyle k=1}$, this reduces to

${\displaystyle [f,g]=fg{f}^{-1}{g}^{-1},}$

which is the usual commutator in ${\displaystyle {\pi }_1(X)}$. This can also be seen by observing that the ${\displaystyle 2}$-cell of the torus ${\displaystyle S^1\times S^1}$ is attached along the commutator in the ${\displaystyle 1}$-skeleton ${\displaystyle S^1\vee S^1}$.

Whitehead products on H-spaces

For a path connected H-space, all the Whitehead products on ${\displaystyle {\pi }_{\ast }(X)}$ 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 ${\displaystyle \alpha \in {\pi }_i(X)}$, ${\displaystyle \beta \in {\pi }_j(X)}$ lie in the kernel of the suspension homomorphism ${\displaystyle \mathrm{\Sigma }:{\pi }_{i+j-1}(X)\to {\pi }_{i+j}(\mathrm{\Sigma }X)}$

Examples

• ${\displaystyle [{\mathrm{i}\mathrm{d}}_{S^2},{\mathrm{i}\mathrm{d}}_{S^2}]=2\cdot \eta \in {\pi }_3(S^2)}$, where ${\displaystyle \eta :S^3\to S^2}$ is the Hopf map.

This can be shown by observing that the Hopf invariant defines an isomorphism ${\displaystyle {\pi }_3(S^2)\cong \mathbb{Z}}$ and explicitly calculating the cohomology ring of the cofibre of a map representing ${\displaystyle [{\mathrm{i}\mathrm{d}}_{S^2},{\mathrm{i}\mathrm{d}}_{S^2}]}$. 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.