= Whitehead square
{c}
{title2=$[\alpha,\alpha]$}
The Whitehead square of a class $\alpha\in\pi_mY$ 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.
Back to article page
