The unstable homotopy groups of 2-cell complexes

Abstract

In this paper, we develop the new method, initiated by B. Gray (1972), to compute the unstable homotopy groups of the mapping cone, especially for $2$-cell complex $X=S^m{\cup }_{\alpha }{e}^n$. By Gray's work mentioned above or the traditional method given by I.M.James (1957) which were widely used in previous related work to compute ${\pi }_i(X)$, the dimension $i\le 2n+m-4$. By our method, we can compute ${\pi }_i(X)$ for $i>2n+m-4$. We use this different technique to generalize J.Wu's work, at Mem. of AMS, on homotopy groups of mod $2$ Moore spaces to higher dimensional homotopy groups of mod $2^r$ Moore spaces $P^n(2^r)$ for all $r\ge 1$. This practice shows that the technique given here is a new general method to compute the unstable homotopy groups of CW complexes with higher dimension.