nLab
Whitehead product

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 $[\alpha ,\beta {]}_{\mathrm{Wh}}\in {\pi }_{r+s-1}(X)$. 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 $[\alpha ,\beta {]}_{\mathrm{Wh}}\in [Y\star Z,X]$. Here $S^{\cdot }$ denotes the reduced suspension operation on pointed spaces and $\star$ denotes the join of CW-complexes. Notice that $\mathrm{pt}\star Z=C^{\cdot }(Z)$ and the reduced cone? of a point is $C^{\cdot }(\mathrm{pt})=S^1$. Thus for $Y=Z=\mathrm{pt}$ the generalized Whitehead product reduced to the usual Whitehead product.

See also wikipedia.