On the homotopy groups of 2-cell complexes

0. The homotopy groups of CW complexes are, in general, much more mysterious than the stable homotopy groups. A notable exception is the case of spheres or cases when the unstable Adams spectral sequence can be utilized. The problem is clearest in the case of a 2-cell complex S ∪ e. Very little knowledge of such spaces was known before the work of Cohen, Moore, and Neisendorfer who analyzed the case of the Moore space S ∪pr e for p an odd prime ([CMN]). Their work gave a clear understanding of the kinds of things that can occur, and the depth of their analysis was demonstrated by their determination of the exponent of the homotopy groups of Moore spaces when p > 2. Our purpose is to discuss the homotopy groups of two cell complexes in case the attaching map is an arbitrary element in an even stem. In some cases we will have results as strong as those in [CMN]. Throughout this paper all spaces will be localized at a prime p > 2. Let θ : S2n−1 −→ S2m−1 and write P = S2m−1 ∪θ e, P r = Sr−2nP for r ≥ 2n, and σ = 2n− 2m+ 1. In section 1, we deal with quite a general decomposition theorem for spaces of the form ΩS(X ∪φ e2n−1). There are two applications