$\mathcal{E}_\infty$ ring spectra and elements of Hopf invariant $1$

TL;DR

This paper investigates properties of certain $ ext{E}_ olinebreak_ olinebreak ext{infty}$ ring spectra related to Hopf invariant 1 elements, focusing on their homology and potential module spectrum structures over known ring spectra.

Contribution

It characterizes the homology of these spectra as extended comodule algebras over the dual Steenrod algebra and explores their possible relations to module spectra over $H ext{Z}$, $kO$, or $ ext{tmf}$.

Findings

Homology of spectra are extended comodule algebras over the dual Steenrod algebra.

Algebra retracts are identified within the homology structures.

Potential connections to module spectra over $H ext{Z}$, $kO$, or $ ext{tmf}$ are suggested.

Abstract

The $2$-primary Hopf invariant $1$ elements in the stable homotopy groups of spheres form the most accessible family of elements. In this paper we explore some properties of the $\mathcal{E}_\infty$ ring spectra obtained from certain iterated mapping cones by applying the free algebra functor. In fact, these are equivalent to Thom spectra over infinite loop spaces related to the classifying spaces $B\mathrm{SO},\,B\mathrm{Spin},\,B\mathrm{String}$. We show that the homology of these Thom spectra are all extended comodule algebras of the form $\mathcal{A}_*\square_{\mathcal{A}(r)_*}P_*$ over the dual Steenrod algebra $\mathcal{A}_*$ with $\mathcal{A}_*\square_{\mathcal{A}(r)_*}\mathbb{F}_2$ as an algebra retract. This suggests that these spectra might be wedges of module spectra over the ring spectra $H\mathbb{Z}$, $k\mathrm{O}$ or $\mathrm{tmf}$, however apart from the first case, we have no concrete results on this.