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One of the most significant modern developments in algebraic topology has been the discovery of periodic phenomena in stable homotopy. This began with Adams’ work on the J homomorphism [Ada66] and was later continued by L. Smith [Smi70] and H. Toda [Tod71]. Let p be a prime and let V (0)k denote the cofiber of the degree p map on S. Adams and Toda showed that for k sufficiently large there is a nonnilpotent map ΣV (0)k α −→ V (0)k, where q = 2(p− 1) if p is odd and q = 8 if p = 2. Let α denote the composite ΣV (0)k −→ · · · −→ ΣV (0)k Σα −→ ΣV (0)k α −→ V (0)k. Then by nonnilpotent, we mean that no α is nullhomotopic. By including the bottom cell of ΣV (0)k and projecting to the top cell of V (0)k, we obtain a family of elements, also denoted αt, in the homotopy of spheres: S −→ ΣV (0)k α −→ V (0)k −→ S.

The Smith-Toda complexV((p+ 1)∕2) does not exist