The relative James construction and its application to homotopy groups

In this paper, we develop the new method to compute the homotopy groups of the mapping cone Cf=Y₅CX beyond the metastable range by analysing the homotopy of the n-th filtration of the relative James construction J (X, A) for CW-pair Ai X, defined by B. Gray, which is homotopy equivalent to the homotopy fiber of the pinch map X₈CA A. As an application, we compute the 5 and 6-dim unstable homotopy groups of 3-dimensional mod 2ʳ Moore spaces for all positive integers r.