3-primary exponents

Abstract

The purpose of this paper is to show that 3n annihilates the 3-primary component of the homotopy groups of the 2n + 1-dimensional sphere. In the terminology of (2) and (3), S2n+1 has exponent 3n at 3.In fact, a stronger result is proved. Localize at 3 and let Ω2nS2n + 1〈2n + 1〉 denote the 2n-fold loop space of the 2n + 1-connected cover of S2n+1. Then Ω2nS2n + 1〈2n + 1〉 has a null homotopic 3n-th power map.