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The $2$-primary Hopf invariant $1$ elements in the stable homotopy groups ofspheres form the most accessible family of elements. In this paper we exploresome properties of the $\mathcal{E}_\infty$ ring spectra obtained from certainiterated mapping cones by applying the free algebra functor. In fact, these areequivalent to Thom spectra over infinite loop spaces related to the classifyingspaces $B\mathrm{SO},\,B\mathrm{Spin},\,B\mathrm{String}$. We show that thehomology 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 $\mathcal{A}_*\square_{\mathcal{A}(r)_*}\mathbb{F}_2$ asan algebra retract. This suggests that these spectra might be wedges of modulespectra 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.

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