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 (Whitehead 1941).

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 Uehara & Massey (1957) 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[1] has only trivial Whitehead products, hence strict groupoids can never model the homotopy types of spheres, such as .[2]

See also

References

  1. Brown, Ronald; Higgins, Philip J. (1981). "The equivalence of ∞-groupoids and crossed complexes". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 22 (4): 371–386.
  2. Simpson, Carlos (1998-10-09). "Homotopy types of strict 3-groupoids". arXiv:math/9810059.
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