Whitehead product

The Whitehead product is a super 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

$S^k \vee S^l$ ;

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

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

π k + l - 1 (X).

[edit] Properties

The Whitehead product is bilinear, super skew-symmetric, and satisfies the super Jacobi relation, and is thus a super quasi-Lie algebra.

If $f \in \pi_1(X)$ , then the Whitehead bracket is related to the usual conjugation action of $π 1$ on $π k$ by

[f, g] = g f - g

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

[f, g] = f g f - 1 g - 1

which is the usual commutator.

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

[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.