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).


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 sum

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 \,


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

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


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.


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.

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.

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