On the odd primary cohomology of higher projective planes (Q1919921)

Let \(C_{n,p}\) \((\supset X)\) denote the \(p\)th filtration part of the May approximation \(C_nX\) to the iterated loop space \(\Omega^n \Sigma^nX\). Kuhn, Slack and Williams call \(X\) to be an \(H^n_p\)-space if there exists a homotopy retraction \(\Theta_n : C_{n,p} \to X\). Such spaces form a category \({\mathcal H}^n_p\) and the functor \(\Omega^n : {\mathcal H}^0_p \to {\mathcal H}^0_p\) admits a left adjoint \(P^n_p : {\mathcal H}^n_p \to {\mathcal H}^0_n\) constructed as follows: Let \(\varepsilon_n : \Sigma^n C_{n,p} X \to\Sigma^nX\) be the adjoint of \(C_{n,p} X \subset C_nX \to \Omega^n \Sigma^n X\) and let \(\widetilde C_{n,p}X\) denote the cofiber of \(X \subset C_{n,p}\). Then \(\Sigma^n \Theta_n - \varepsilon_n\) can be extended to \(h: \Sigma^n \widetilde C_{n,p} \to \Sigma^n X\). The higher projective plane \(P^n_pX\) is defined to be the cofiber of \(h\) with the cofibration sequence \[ \Sigma^n \widetilde C_{n,p} X @>h>> \Sigma^n X @>i>> P^n_p @>j>> \Sigma^{n+1} \widetilde C_{n,p}. \] With these situations the authors show, as the main theorem, that the reduced powers of \(\overline x \in H^* (P^n_pX, {\mathfrak F}_p)\) can be expressed, up to a unit factor in \({\mathfrak F}_p\), as the \(j^*\) image of the iterated cohomology suspension \(\sigma^{n+1}\) of \(x\) applied by the dual external Dyer-Lashof operation with \(i^*(\overline x) = \sigma^nx\).