Whitehead product
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The Whitehead product is a super quasi-Lie algebra structure on the homotopy groups of a space.
Given elements , the Whitehead bracket
is defined as follows:
The product can be obtained by attaching a (k + l)-cell to the wedge product
- ;
the attaching map is a map
- .
Represent f and g by maps
and
- ,
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
- πk + l − 1(X).
[edit] Properties
The Whitehead product is bilinear, super skew-symmetric, and satisfies the super Jacobi relation, and is thus a super quasi-Lie algebra.
If , then the Whitehead bracket is related to the usual conjugation action of π1 on πk by
- [f,g] = gf − g,
where gf denotes the conjugation of g by f. For k = 1, this reduces to
- [f,g] = fgf − 1g − 1,
which is the usual commutator.
The relevant MSC code is: 55Q15, Whitehead products and generalizations.