$E_{\infty}$ obstruction theory

The space of $E_{\infty}$ structures on a simplicial operad $\mathcal{C}$ 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 $\Gamma$-module $\pi_{*} \mathcal{C}$; the fringe contains an obstruction theory for the existence of $E_{\infty}$ structures on $\mathcal{C}$. This formulation is very flexible: applications extend beyond structures on classical ring spectra to examples in motivic homotopy theory.

Keywords

operad, $E_{\infty}$ structure, Bousfield–Kan spectral sequence

2010 Mathematics Subject Classification

18D50, 55P43, 55P48, 55S35

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Received 22 November 2016

Received revised 25 August 2017

Published 24 January 2018