The Whitehead squares \([l_{2^ i-1}, l_{2^ i-1}]\in \pi_{2^{i+1}-3} (S^{2^ i-1})\) are projective (Q1804908)

A homotopy element in \(\pi_n (X)\) is said to be projective if it can be represented by a map \(f\) that factors through the real projective space \(P^n\). This means \(f = \overline f \sigma : S^n @>\sigma>> P^n @> \overline f >> X\) for some \(\overline f\), where \(\sigma\) is the double covering map. In this paper the following result is proved: Theorem. The Whitehead product \([\iota_n, \iota_n] \in \pi_{2n - 1} (S^n)\) is projective for \(n = 2^i - 1\) and \(i \geq 4\). Discussing the literature the author shows that by this result all \(n\) with projective \([\iota_n, \iota_n]\) are now known. The further values are \(n = 2, 4, 8\), and by trivial reasons \(n = 1, 3, 7\), because in these last 3 cases \([\iota_n, \iota_n] = 0\). The proof of the Theorem uses the \(k\)th 2-primary Brown-Gitler spectrum \(B(k)\) and the Adams spectral sequences for \(\pi^S_* (B(k) \wedge Y)\).