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.

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

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Whitehead product

Definition

Grading

Properties

See also

References