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Abstract

Localize spaces at 2. Let $f\in\pi_{n+k}(S^{n})$, ($n\geq2$ and $k\geq0$) and let $F$ be the homotopy fibre of the pinch map $\Sigma C_{f}=S^{n+1}\cup e^{n+k+2} \rightarrow S^{n+k+2}$, (in the case $k=0$, further assume $\Sigma C_{f}\not\Simeq*$). As a type of Gray's relative James constructions, it is known that $ F$ has a CW decomposition $F=(S^{n+1}\cup_{\af} e^{2n+k+2})\cup_{\beta} e^{3n+2k+3}\cup \cdots$, where $\af=[1_{S^{n+1}},\s f]\in\pi_{n+k+1}(S^{n})$ is the Whitehead product and $\beta$ is a higher Whitehead product. However, higher Whitehead products are hard to control in general. In this paper, by use of Selick-Wu's $A^{\text{min}}$-theory together with the machinery of Eilenberg-Moore spectral sequences, we show that in the 2-local case, the attaching map $\beta\in \pi_{3n+2k+2}(S^{n+1}\cup_{\af} e^{2n+k+2})$ is decomposed to be $\s^{2n+k+2}f$ composing with certain inclusion map. By this, we are able to calculate the higher unstable homotopy groups $\pi_{*}(\Sigma C_{f})$ by the homoptopy fibration $F\rightarrow \Sigma C_{f} \rightarrow S^{n+k+2}$ in the range requiring the 3-cell skeleton of $F$ if the involved homotopy groups of spheres are known. As an application, we determine the homotopy group $\pi_{18}(\Sigma^{3}\mathbb{C}P^{2}),$ a higher unstable homotopy group.