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 $C_f=Y{\cup }_{f}CX$ beyond the metastable range by analysing the homotopy of the $n$-th filtration of the relative James construction $J(X,A)$ for CW-pair $AX$, defined by B. Gray, which is homotopy equivalent to the homotopy fiber of the pinch map $X{\cup }_{i}CA\to \mathrm{\Sigma }A$. As an application, we compute the 5 and 6-dim unstable homotopy groups of 3-dimensional mod $2^r$ Moore spaces for all positive integers $r$.