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

Beben, Piotr; Theriault, Stephen

doi:10.1090/S0002-9939-2012-11337-1

The Kahn-Priddy Theorem and the homotopy of the three-sphere
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Proc. Amer. Math. Soc. 141 (2013), 711-723 Request permission

Abstract:

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

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