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.

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 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 $$\pi_{3}(S^{2}) \cong \Z$$ 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.