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

Let $f\inπ_{n+k}(S^{n})$, ($n\geq2$ and $k\geq0$) and let $F$ be the homotopy fibre of the pinch map $ΣC_{f}=S^{n+1}\cup e^{n+k+2} \rightarrow S^{n+k+2}$; (further suppose that $ΣC_{f}\not\simeq*$ if $k=0$). Assume that, via the homoptopy fibration $F\rightarrow ΣC_{f} \rightarrow S^{n+k+2}$, we wish to compute the $p$-primary component of $π_{*}(ΣC_{f})$ which is possibly far unstable, where the prime $p\neq3$. As a type of Gray's relative James constructions, $ F$ has a CW decomposition $F=(S^{n+1}\cup_α e^{2n+k+2})\cup_β e^{3n+2k+3}\cup \cdots$, where $α=[1_{S^{n+1}},Σf]$ is the Whitehead product and $β$ is a higher Whitehead product. However, higher Whitehead products are intricate to manage. In this paper, by use of Selick-Wu's $\mathrm{A}^{\mathrm{min}}$-theory and the machinery of the Eilenberg-Moore spectral sequence, we show that in the 2-local case, the attaching map $β$ is decomposed to be $Σ^{2n+k+2}f$ composing with a certain inclusion; in the $p$-local case, where $p\geq5$ and $n+k$ is odd, $β$ admits a similar decomposition. In other words, the attaching map $Σf$ manifests a ``periodicity'' phenomenon. Therefore, we are able to calculate $π_{*}(ΣC_{f})\otimes \mathbb{Z}_{(p)}$ in the range requiring the 3-cell skeleton of $F$. As an application, we determine the 2-primary component of $π_{18}(Σ^{3}\mathbb{C}P^{2})$.