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 Template:Harv.

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

$S^{k+l-1}\stackrel{\varphi }{⟶}{S}^k\vee S^l.$

Represent $f$ and $g$ by maps

$f:S^k\to X$

and

$g:S^l\to X,$

then compose their wedge with the attaching map, as

$S^{k+l-1}\stackrel{\varphi }{⟶}{S}^k\vee S^l\stackrel{f\vee g}{⟶}X.$

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

Sometimes the homotopy groups of a space, together with the Whitehead product operation are called a graded quasi-Lie algebra; this is proven in Template:Harvtxt 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

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

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 $\mathrm{\Sigma }:{\pi }_{i+j-1}(X)\to {\pi }_{i+j}(\mathrm{\Sigma }X)$

Examples

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

See also

• Generalised Whitehead product
• Massey product
• Toda bracket

References

• Template:Citation
• Template:Citation
• Template:Citation
• Template:Cite book