A "Periodicity" Phenomenon of the Attaching Map of the Suspended Two-Cell Complex

In this paper, we determine the 3-cell skeleton of $F$, where $F$ is the homotopy fiber of the canonical pinch map from a suspension of a simply-connected 2-cell complex onto a sphere. The main result is stated $p$-locally: for $p=2$, and for $p\ge 5$ under an additional assumption. The proof is based on Selick-Wu’s ${\mathrm{A}}^{\mathrm{m}\mathrm{i}\mathrm{n}}$-theory and the machinery of the Eilenberg-Moore spectral sequence. As an application, we compute the 2-primary component of ${\pi }_{18}({\mathrm{\Sigma }}^3\mathbb{C}{P}^2)$, a homotopy group outside the metastable range.