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 Bmathrm {SO},,Bmathrm {Spin},,Bmathrm {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 Hmathbb {Z}, kmathrm {O} or mathrm {tmf}; however, apart from the first case, we have no concrete results on this.
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