Whitehead product

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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[edit]

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[edit]

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[edit]

The Whitehead product is bilinear, graded-symmetric, and satisfies the graded Jacobi identity, and is thus a graded quasi-Lie algebra; this is proven in Uehara & Massey (1957) via the Massey triple product.

Relation to the action of [edit]

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[edit]

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 group of H-spaces are abelian, and that H-spaces are simple.

Suspension[edit]

All Whitehead products of classes , lie in the kernel of the suspension homomorphism

Examples[edit]

  • , 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 .

See also[edit]

References[edit]

  • Whitehead, J. H. C. (April 1941), "On adding relations to homotopy groups", Annals of Mathematics, 2, 42 (2): 409–428, doi:10.2307/1968907, JSTOR 1968907
  • 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, MR 0091473
  • Whitehead, George W. (July 1946), "On products in homotopy groups", Annals of Mathematics, 2, 47 (3): 460–475, doi:10.2307/1969085, JSTOR 1969085
  • Whitehead, George W. "X.7 The Whitehead Product". Elements of homotopy theory. Springer-Verlag. pp. 472–487. ISBN 978-0387903361.