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

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.