Rigidity of the $K(1)$-local stable homotopy category

TL;DR

This paper proves the rigidity of the $K(1)$-local stable homotopy category at prime 2, showing that its triangulated structure uniquely determines its higher homotopy information.

Contribution

It establishes a new case of rigidity in stable homotopy theory, demonstrating that the $K(1)$-local category's triangulated structure suffices to recover all homotopical data.

Findings

Rigidity of the $K(1)$-local stable homotopy category at p=2 established.

Higher homotopy information is recoverable from the triangulated structure.

Only a few stable model categories are known to have this rigidity property.

Abstract

We investigate a new case of rigidity in stable homotopy theory which is the rigidity of the $K(1)$-local stable homotopy category $\mathrm{Ho}(L_{K(1)}\mathrm{Sp})$ at $p=2$. In other words, we show that recovering higher homotopy information by just looking at the triangulated structure of $\mathrm{Ho}(L_{K(1)}\mathrm{Sp})$ is possible, which is a property that only few interesting stable model categories are known to possess.