The Kahn–Priddy theorem

Abstract

Let ΦSr(X) be the stable homotopy groupwhere SnX means the n-fold suspension of X. For example, the groups ΦSr(S0) are the stable homotopy groups of spheres. Letbe the ‘infinite-dimensional’ orthogonal group. Then topologists are familiar with the ‘stable J-homomorphism’G. W. Whitehead observed that J factors through an ‘even more stable’ J-homomorphismhe conjectured that J′ is epi (for r > 0).