Given elements , the Whitehead bracket
is defined as follows:
The product can be obtained by attaching a -cell to the wedge sum
the attaching map is a map
Represent and by maps
then compose their wedge with the attaching map, as
The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of
Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so has degree ; equivalently, (setting L to be the graded quasi-Lie algebra). Thus 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 , then the Whitehead bracket is related to the usual conjugation action of on by
where denotes the conjugation of by . For , this reduces to
which is the usual commutator.
The relevant MSC code is: 55Q15, Whitehead products and generalizations.
- 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