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# Homology, Homotopy and Applications

## Volume 20 (2018)

### Number 1

### $E_{\infty}$ obstruction theory

Pages: 155 – 184

DOI: https://dx.doi.org/10.4310/HHA.2018.v20.n1.a10

#### Author

#### Abstract

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

Received 22 November 2016

Received revised 25 August 2017

Published 24 January 2018