Whitehead product

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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 \times S^l can be obtained by attaching a (k + l)-cell to the wedge product

S^k \vee S^l;

the attaching map is a map

S^{k+l-1} \to S^k \vee S^l.

Represent f and g by maps

f\colon S^k \to X

and

g\colon S^l \to X,

then compose their wedge with the attaching map, as

S^{k+l-1} \to S^k \vee S^l \to 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

πk + l − 1(X).

[edit] Grading

Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so πk(X) has degree (k − 1); equivalently, Lk = πk + 1(X) (setting L to be the graded quasi-Lie algebra). Thus L0 = π1(X) acts on each graded component.

[edit] 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 π1 on πk by

[f,g] = gf − g,

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

[f,g] = fgf − 1g − 1,

which is the usual commutator.

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

[edit] References

  • [1] J. H. C. Whitehead, On adding relations to homotopy groups, Annals of Mathematics, 2nd Ser., Vol. 42, No. 2. (Apr., 1941), pp. 409 –428.
  • [2] 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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