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 ${\displaystyle f\in \pi _{k}(X),g\in \pi _{l}(X)}$, the Whitehead bracket

${\displaystyle [f,g]\in \pi _{k+l-1}(X)\,}$

is defined as follows:

The product ${\displaystyle S^{k}\times S^{l}}$ can be obtained by attaching a ${\displaystyle (k+l)}$-cell to the wedge sum

${\displaystyle S^{k}\vee S^{l}}$;

the attaching map is a map

${\displaystyle S^{k+l-1}\to S^{k}\vee S^{l}.\,}$

Represent ${\displaystyle f}$ and ${\displaystyle g}$ by maps

${\displaystyle f\colon S^{k}\to X\,}$

and

${\displaystyle g\colon S^{l}\to X,\,}$

then compose their wedge with the attaching map, as

${\displaystyle S^{k+l-1}\to S^{k}\vee S^{l}\to X\,}$

The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of

${\displaystyle \pi _{k+l-1}(X).\,}$

Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so ${\displaystyle \pi _{k}(X)}$ has degree ${\displaystyle (k-1)}$; equivalently, ${\displaystyle L_{k}=\pi _{k+1}(X)}$ (setting L to be the graded quasi-Lie algebra). Thus ${\displaystyle L_{0}=\pi _{1}(X)}$ 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 ${\displaystyle f\in \pi _{1}(X)}$, then the Whitehead bracket is related to the usual conjugation action of ${\displaystyle \pi _{1}}$ on ${\displaystyle \pi _{k}}$ by

${\displaystyle [f,g]=g^{f}-g,\,}$

where ${\displaystyle g^{f}}$ denotes the conjugation of ${\displaystyle g}$ by ${\displaystyle f}$. For ${\displaystyle k=1}$, this reduces to

${\displaystyle [f,g]=fgf^{-1}g^{-1},\,}$

which is the usual commutator.

The relevant MSC code is: 55Q15, Whitehead products and generalizations.