$E_{\infty}$ obstruction theory

Abstract

The space of E ∞ structures on a simplicial operad C is the limit of a tower of fibrations, so its homotopy is the abutment of a Bousfield-Kan fringed spectral sequence. The spectral sequence begins (under mild restrictions) with the stable cohomotopy of the right Γ-module π * C; the fringe contains an obstruction theory for the existence of E ∞ structures on C. This formulation is very flexible: applications extend beyond structures on classical ring spectra to examples in motivic homotopy theory.