The Whitehead square of a lift of the Hopf map to a mod 2 Moore space. (Q1411727)

The present paper is a continuation of [J. Math. Soc. Japan 52, 515--533 (2000; Zbl 0960.55006)]. Writing \(n_n\in \pi_{n+1}(S^n)\), \(n\geq 2\), for the suspension of the Hopf map, it is known that \(n_3\) lifts to the Moore space \(M(Z_2,n-1)\), with projection map \(M(Z_2,n-1)\to S^n\). Calling the lift \(\hat n_3\), the authors write \(\hat n_n =\sum^{n-3}\hat n_3\), \(n\geq 3\). This paper studies the Whitehead square \([\hat n_n,\hat n_n]\). The authors calculate the order when \(n\) is odd, and when \(n=2m\), \(m\neq 2^i-1\). The methods include a relation with the \(J\)-homomorphism and relative Whitehead products. The details are much too complex to be included here.