Whitehead product ($[\alpha ,\beta ]\in {\pi }_{p+q-1}Y$)

The geometric Whitehead product of based maps $\alpha :S^p\to Y$, $\beta :S^q\to Y$ is their wedge sum map composed with the attaching map $S^{p+q-1}\to S^p\vee S^q$ of the top cell of $S^p\times S^q$. With the usual geometric signs it satisfies $[\alpha ,\beta {]}_W=(-1{)}^{pq}[\beta ,\alpha {]}_W$. Regrading ${\pi }_{p}Y$ to degree $p-1$ and setting $\{\alpha ,\beta \}=(-1{)}^p[\alpha ,\beta {]}_W$ gives the graded Lie bracket transported from the Samelson product on the loop space. In degrees $p,q\ge 2$ it is bilinear; fundamental-group inputs instead record commutators and the action of ${\pi }_1$ on higher homotopy.

The Whitehead square of a class $\alpha \in {\pi }_{m}Y$ is $[\alpha ,\alpha ]\in {\pi }_{2m-1}Y$. For the identity class of an even-dimensional sphere, the Hopf invariant is two with the standard product-cell orientation: folding the two sphere summands makes the middle cohomology class of the mapping cone pull back to $a+b$ on $S^m\times S^m$, and $(a+b{)}^2=2ab$. For odd $m$, graded symmetry makes the square two-torsion, hence rationally zero.