E_∞ ring spectra and elements of Hopf invariant 1

2015/03/19 by Baker, Andrew
#55P42 #55P43 (Primary) #57T05 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1503.05902

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 E_∞ 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 BSO, BSpin, BString. We show that the homology of these Thom spectra are all extended comodule algebras of the form A_*\square_{A(r)_*}P_* over the dual Steenrod algebra A_* with A_*\square_{A(r)_*}\mathbbF₂ as an algebra retract. This suggests that these spectra might be wedges of module spectra over the ring spectra Hℤ, kO or tmf, however apart from the first case, we have no concrete results on this.

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