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

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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

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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

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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  

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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

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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

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

Examples

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  •  , 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.

See also

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References

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  • 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. (1978). "X.7 The Whitehead Product". Elements of homotopy theory. Springer-Verlag. pp. 472–487. ISBN 978-0387903361.