H-infinity is not E-infinity

TL;DR

This paper presents a spectrum with an H-infinity structure that cannot be enhanced to an E-infinity structure, showing limitations in the relationship between these algebraic structures in spectra.

Contribution

It provides a concrete example demonstrating that not all H-infinity ring spectra originate from E-infinity ring spectra, clarifying the distinction between these structures.

Findings

An explicit spectrum with H-infinity but not E-infinity structure.

H-infinity structures do not always rigidify to E_3 structures.

Counterexample derived from the transfer conjecture by Kraines and Lada.

Abstract

We provide an example of a spectrum over S^0 with an H_\infty structure which does not rigidify to an E_3 structure. It follows that in the category of spectra over S^0 not every H_\infty ring spectrum comes from an underlying E_\infty ring spectrum. After comparing definitions, we obtain this example by applying \Sigma^\infty_+ to the counterexample to the transfer conjecture constructed by Kraines and Lada.