Characteristics for $E_\infty$ ring spectra

TL;DR

This paper introduces a new concept of characteristic for connective p-local E-infinity ring spectra, exploring its properties and examples, and proposing conjectures linked to deep questions in stable homotopy theory.

Contribution

It defines the notion of characteristic for connective p-local E-infinity ring spectra and investigates examples involving Hopf invariant 1 elements, connecting to complex stable homotopy questions.

Findings

Examples from Hopf invariant 1 elements are analyzed.

Conjectures relate characteristics to deep stable homotopy problems.

Basic properties of the new characteristic notion are established.

Abstract

We introduce a notion of characteristic for connective $p$-local $E_\infty$ ring spectra and study some basic properties. Apart from examples already pointed out by Markus Szymik, we investigate some examples built from Hopf invariant $1$ elements in the stable homotopy groups of spheres and make some conjectures about spectra for which they may be characteristics; these appear to involve hard questions in stable homotopy theory.