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 , the Whitehead bracket 

is defined as follows:

The product  can be obtained by attaching a -cell to the wedge sum

;

the attaching map is a map 

Represent  and  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  has degree ; equivalently,  (setting L to be the graded quasi-Lie algebra). Thus  acts on each graded component. Properties

The Whitehead product satisfies the following properties:

 Bilinearity. Graded Symmetry. Graded Jacobi identity. 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  

If , then the Whitehead bracket is related to the usual action of  on  by

where  denotes the conjugation of  by . For , this reduces to

which is the usual commutator in . This can also be seen by observing that the -cell of the torus  is attached along the commutator in the -skeleton . Whitehead products on H-spaces 

For a path connected H-space, all the Whitehead products on  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 ,  lie in the kernel of the suspension homomorphism

Examples

 , where  is the Hopf map. This can be shown by observing that the Hopf invariant defines an isomorphism  and explicitly calculating the cohomology ring of the cofibre of a map representing . 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. Applications to ∞-groupoids 
Recall that an ∞-groupoid  is an -category generalization of groupoids which is conjectured to encode the data of the homotopy type of  in an algebraic formalism. The objects are the points in the space , 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 ∞-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 . See also
 Generalised Whitehead product
 Massey product
 Toda bracket

References

 
  
 
 

Homotopy theory
Lie algebras