On the metastable homotopy of \(\bmod 2\) Moore spaces (Q2630681)

Let \(P^n(2)=\Sigma^{n-2}\mathbb{R}\mathrm{P}^2\) be the \(n\) dimensional mod \(2\) Moore space with \(n\geq 3\). It is well-known that the metastable homotopy of \(P^n(2)\) has an exponent dividing \(8\) and this leads to the natural question whether the metastable homotopy of \(P^n(2)\) has an exponent \(4\). It is also known that it has an exponent \(8\) when \(n\equiv 2, 3\) mod \(4\). In this paper, the authors consider this question for the case \(n\equiv 0\) mod \(4\), and they prove that the homotopy group of the double loop space \(\Omega^2P^{4n}(2)\) has the multiplicative exponent \(4\) below the range of \(4\) times connectivity by using the Cohen group for displaying the explicit obstructions to the \(4\)th power map on \(\Omega P^n(2)\) and shuffle relations with Hopf invariants on general configuration spaces.