nLab
Whitehead product

Idea

The Whitehead product is a bilinear operation on the elements of the homotopy groups of a pointed CW-complex $X$.

More specifically, it takes as input $\alpha\in\pi_r(X)$ and $\beta\in\pi_s(X)$ and produces $[\alpha,\beta]_{Wh}\in\pi_{r+s-1}(X)$. The operation satisfies a graded Jacobi identity (the conventions on the signs are not uniform in the literature).

There is also a generalized Whitehead product where we can take more general homotopy classes (continuous maps up to homotopy) $\alpha\in [S^\cdot Y,X]$ and $\beta\in [S^\cdot Z,X]$ to produce a class $[\alpha,\beta]_{Wh}\in[Y\star Z,X]$. Here $S^\cdot$ denotes the reduced suspension operation on pointed spaces and $\star$ denotes the join of CW-complexes. Notice that $pt\star Z = C^\cdot(Z)$ and the reduced cone? of a point is $C^\cdot(pt)=S^1$. Thus for $Y=Z=pt$ the generalized Whitehead product reduced to the usual Whitehead product.

The Whitehead products form one of the primary homotopy operations and in fact with composition operations and $\pi_1$-actions generate all such operations. This is related to the definition of Pi-algebra.

In the context of simplicial groups, representing connected homotopy types, there is a formula for the Whitehead product in terms of a Samelson product?, which in turn is derived from a shuffle product which is a sort of non-commutative version of the Eilenberg-Zilber map. These simplicial formulae come from an analysis of the structure of the product of simplices. (The formula for the Whitehead product is due to Dan Kan and can be found in the old survey article of Ed Curtis. The proof that it works was never published.)

References

Named after J. H. C. Whitehead.

See also

• Wikipedia Whitehead product