Whitehead product

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).

Definition

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

$f\colon S^k \to X \,$

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

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

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; 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.

See also

• Generalised Whitehead product
• Massey product
• Toda bracket

References

• Uehara, Hiroshi; Massey, W. 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, 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