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 formula_1, the Whitehead bracket 
is defined as follows:
The product formula_3 can be obtained by attaching a formula_4-cell to the wedge sum
the attaching map is a map 
Represent formula_7 and formula_8 by maps
and
then compose their wedge with the attaching map, as
The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of 
Grading.
Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so formula_13 has degree formula_14; equivalently, formula_15 (setting "L" to be the graded quasi-Lie algebra). Thus formula_16 acts on each graded component.
Properties.
The Whitehead product satisfies the following properties:
Sometimes the homotopy groups of a space, together with the Whitehead product operation are called a [[Graded Lie algebra|graded]] [[quasi-Lie algebra]]; this is proven in via the [[Massey product|Massey triple product]].
Relation to the action of formula_20.
If formula_21, then the Whitehead bracket is related to the usual action of formula_22 on formula_23 by
where formula_25 denotes the [[Inner automorphism|conjugation]] of formula_8 by formula_7. 
For formula_28, this reduces to
which is the usual [[Commutator#Group theory|commutator]] in formula_30. This can also be seen by observing that the formula_31-cell of the torus formula_32 is attached along the commutator in the formula_33-skeleton formula_34.
Whitehead products on H-spaces.
For a path connected [[H-space]], all the Whitehead products on formula_35 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 space|simple]].
Suspension.
All Whitehead products of classes formula_36, formula_37 lie in the kernel of the [[Suspension (topology)|suspension]] homomorphism formula_38
Examples.
This can be shown by observing that the [[Hopf invariant]] defines an isomorphism formula_41 and explicitly calculating the cohomology ring of the cofibre of a map representing formula_42. Using the [[Thom space|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]]. 
Applications to ∞-groupoids.
Recall that an ∞-groupoid formula_43 is an [[Infinity category|formula_44-category]] generalization of [[Groupoid|groupoids]] which is conjectured to encode the data of the [[homotopy type]] of formula_45 in an algebraic formalism. The objects are the points in the space formula_45, morphisms are homotopy classes of paths between points, and higher morphisms are higher homotopies of those points.
The existence of the Whitehead product is the main reason why defining a notion of [[∞-groupoid|∞-groupoids]] is such a demanding task. It was shown that any strict ∞-groupoid has only trivial Whitehead products, hence strict groupoids can never model the homotopy types of spheres, such as formula_47.
References.
[[Category:Homotopy theory]]
[[Category:Lie algebras]]