Atoms in the p-localization of stable homotopy category

TL;DR

This paper classifies indecomposable objects in certain p-localized subcategories of the stable homotopy category for low dimensions, revealing a wild classification problem beyond a specific threshold.

Contribution

It provides a classification of atoms in p-localized subcategories of the stable homotopy category for dimensions up to 4(p-1), and demonstrates the complexity becomes wild afterwards.

Findings

Classified atoms in S_p^n for n ≤ 4(p-1).

Established wildness of classification for n > 4(p-1).

Applied triangulated categories and matrix problems techniques.

Abstract

We study $p$-localizations, where $p$ is an odd prime, of the full subcategories $S^n$ of stable homotopy category consisting of CW-complexes having cells in $n$ successive dimensions. Using the technique of triangulated categories and matrix problems we classify atoms (indecomposable objects) in $S_p^n$ for $n\le 4(p-1)$ and show that for $n>4(p-1)$ such classification is wild in the sense of the representation theory.