Whitehead product

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. [f,f]\neq 0.) EDIT: See the example below: The Whitehead product of the identity of S^2 with itself is a nonzero multiple of the Hopf map, and the Hopf map has infite order, so this is nontrivial --

Relation to the action of $\backslash pi\_\{1\}$

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.

References

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