Kahn-Priddy theorems via the norm

TL;DR

This paper offers a modern equivariant homotopy theory approach to the Kahn-Priddy theorem, providing new proofs and extending results to various homotopy theories.

Contribution

It introduces a unified proof technique using multiplicative norms and the Adams isomorphism, applicable across multiple homotopy theoretical contexts.

Findings

New Kahn-Priddy theorems in $L_n$ and $L_n^f$-local homotopy theory

Extensions to motivic and synthetic homotopy theories

A simplified proof approach leveraging equivariant methods

Abstract

We revisit the Kahn-Priddy theorem from the perspective of modern equivariant homotopy theory. This allows for a short proof that may be applied in other settings with sufficiently robust analogues of multiplicative norms and the Adams isomorphism. We illustrate this by establishing new Kahn-Priddy theorems in $L_n$ and $L_n^f$-local homotopy theory, motivic homotopy theory, and synthetic homotopy theory.