The $n+3$, $n+4$ dimensional homotopy groups of ${\mathbf{A}}_n^2$-complexes

In this paper, we calculate the $n+3$, $n+4$ dimensional homotopy groups of indecomposable ${\mathbf{A}}_n^2$-complexes after localization at 2. This job is seen as a sequel to P.J. Hilton’s work on the $n+1,n+2$ dimensional homotopy groups of ${\mathbf{A}}_n^2$-complexes (1950-1951). The main technique used is analysing the homotopy property of $J(X,A)$, defined by B. Gray for a CW-pair $(X,A)$, which is homotopy equivalent to the homotopy fibre of the pinch map $X\cup CA\to \mathrm{\Sigma }A$. By the way, the results of these homotopy groups have been used to make progress on recent popular topic about the homotopy decomposition of the (multiple) suspension of oriented closed manifolds.