= Whitehead product
{c}
{title2=$[\alpha,\beta]\in\pi_{p+q-1}Y$}
{wiki}
= Whitehead bracket
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{synonym}
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 $\pi_pY$ 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 $p,q\ge2$ it is bilinear; fundamental-group inputs instead record <commutators> and the action of $\pi_1$ on higher <homotopy>.
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