The Kahn-Priddy Theorem and the homotopy of the three-sphere

Abstract

Let p p be an odd prime. The least nontrivial p p -torsion homotopy group of S 3 S^{3} occurs in dimension 2 p 2p and is of order p p . This induces a map f : P 2 p + 1 ( p ) → S 3 f\colon P^{2p+1}(p)\rightarrow S^{3} , where P 2 p + 1 ( p ) P^{2p+1}(p) is a mod- p p Moore space. An important conjecture related to the Kahn-Priddy Theorem is that the double loops on the three-connected cover of f f has a right homotopy inverse. We prove a weaker but still useful property: if X X is the cofiber of f f , then the double loop on the three-connected cover of the inclusion S 3 → X S^{3}\rightarrow X is null homotopic.