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 , 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.

See also[edit]