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:
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
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.
Uehara, Hiroshi;
Massey, William S. (1957), "The Jacobi identity for Whitehead products", Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton, N. J.:
Princeton University Press, pp. 361–377,
MR0091473