The sphere of semiadditive height 1

TL;DR

This paper constructs a lift of the p-complete sphere to a universal semiadditive stable category at height 1, providing a counterexample to a conjecture and offering new insights into stable cohomotopy of Eilenberg--MacLane spaces.

Contribution

It introduces a novel lift of the p-complete sphere to a semiadditive category at height 1, challenging existing conjectures and connecting to classical cohomotopy results.

Findings

Counterexample to the conjecture that tsade-$n$ is equivalent to $ ext{Sp}_{T(n)}$ at height 1.

Provides a conceptual proof of Lee's theorem on stable cohomotopy.

Establishes a new construction linking semiadditive categories and stable homotopy theory.

Abstract

We construct a lift of the $p$-complete sphere to the universal height $1$ higher semiadditive stable $\infty$-category tsade-$1$ of Carmeli--Schlank--Yanovski, providing a counterexample, at height $1$, to their conjecture that the natural functor from tsade-$n$ to $\mathrm{Sp}_{T(n)}$ is an equivalence. We then record some consequences of the construction, including an observation of T. Schlank that this gives a conceptual proof of a classical theorem of Lee on the stable cohomotopy of Eilenberg--MacLane spaces.