A remark on Hopkins' chromatic splitting conjecture

TL;DR

This paper discusses the structure of homotopy groups in localized sphere spectra, clarifying the distribution of certain copies of alZ in the context of Hopkins' chromatic splitting conjecture.

Contribution

It provides a simplified explanation of the distribution pattern of alZ copies in the homotopy groups related to Hopkins' conjecture.

Findings

Clarifies the occurrence of alZ copies in homotopy groups.

Simplifies the complex book-keeping of these copies.

Provides insights into the structure predicted by Hopkins' conjecture.

Abstract

Ravenel proved the remarkable fact that the $K$-theoretic localization $L_K S^0$ of the sphere spectrum has $\mathbb{Q}/\mathbb{Z}$ as homotopy group in dimension -2. Mike Hopkins' chromatic splitting conjecture implies more generally that there are $3^{n-1}$ copies of $(\mathbb{Q}/\mathbb{Z})_p$ in the homotopy groups of the $E(n)$-localization of $S^0$; but where these copies occur can be confusing. We try here to simplify this book-keeping.