An application of the iterated loop space theory to cohomology suspensions (Q788322)

Let X be an n-fold loopspace, \(X=\Omega^ nY\). Let \(\xi_ n: \Sigma^ nX\to Y\) denote the evaluation map (the adjoint of \(l_ X)\). In cohomology, \(\xi_ n\) induces the suspension map \(\tilde H^ i(Y)\to \tilde H^{i-n}(X)\). \textit{R. J. Milgram} [Unstable homotopy from the stable point of view (Lect. Notes Math. 368) (1974; Zbl 0274.55015)] showed that if X is (m-1)-connected \((m>1)\) then the fibre of \(\xi_ n\) is \((3m+n-l)\)-equivalent to the half-smash product \(S^{n- l}\ltimes_{\Sigma_ 2}(X\bigwedge X)\). The author extends Milgram's result to prove a \((4m+n-1)\)-equivalence.