The n+3, n+4 dimensional homotopy groups of Aₙ²-complexes

In this paper, we calculate the n+3, n+4 dimensional homotopy groups of indecomposable Aₙ²-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 Aₙ²-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 CA 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.