Whitehead product

Whitehead product

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 an Annals of Mathematics paper from 1941.

Given elements f in pi_k(X), g in pi_l(X), the Whitehead bracket

: [f,g] in pi_{k+l-1}(X)

is defined as follows:

The product S^k imes S^l can be obtained by attaching a (k+l)-cell to the wedge sum

:S^k vee S^l;

the attaching map is a map

:S^{k+l-1} o S^k vee S^l.

Represent f and g by maps

:fcolon S^k o X

and:gcolon S^l o X,

then compose their wedge with the attaching map, as

:S^{k+l-1} o S^k vee S^l o X

The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of

:pi_{k+l-1}(X).

Grading

Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so pi_k(X) has degree (k-1); equivalently, L_k = pi_{k+1}(X) (setting "L" to be the graded quasi-Lie algebra). Thus L_0 = pi_1(X) acts on each graded component.

Properties

The Whitehead product is bilinear, graded-symmetric, and satisfies the graded Jacobi identity, and is thus a graded quasi-Lie algebra.

If f in pi_1(X), then the Whitehead bracket is related to the usual conjugation action of pi_1 on pi_k by

: [f,g] =g^f-g,

where g^f denotes the conjugation of g by f.For k=1, this reduces to

: [f,g] =fgf^{-1}g^{-1},

which is the usual commutator.

The relevant MSC code is:55Q15, Whitehead products and generalizations.

References

* [http://links.jstor.org/sici?sici=0003-486X%28194104%292%3A42%3A2%3C409%3AOARTHG%3E2.0.CO%3B2-5] J. H. C. Whitehead, "On adding relations to homotopy groups", Annals of Mathematics, 2nd Ser., Vol. 42, No. 2. (Apr., 1941), pp. 409 –428.
* [http://links.jstor.org/sici?sici=0003-486X%28194607%292%3A47%3A3%3C460%3AOPIHG%3E2.0.CO%3B2-T] George W. Whitehead, "On products in homotopy groups", Annals of Mathematics, 2nd Ser., Vol. 47, No. 3. (Jul., 1946), pp. 460 –475.


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