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 Mathematics [Subject Classification|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, so has degree ; equivalently, . Thus 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 quasi-Lie algebra; this is proven in via the Massey triple product.

Relation to the action of \pi_{1}

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

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.