Formality of ${\mathbb{E}}_n$-algebras and cochains on spheres

Abstract

We study the loop and suspension functors on the category of augmented ${\mathbb{E}}_n$-algebras. One application is to the formality of the cochain algebra of the $n$-sphere. We show that it is formal as an ${\mathbb{E}}_n$-algebra, also with coefficients in general commutative ring spectra, but rarely ${\mathbb{E}}_{n+1}$-formal unless the coefficients are rational. Along the way we show that the free functor from operads in spectra to monads in spectra is fully faithful on a nice subcategory of operads which in particular contains the stable ${\mathbb{E}}_n$-operads for finite $n$. We use this to interpret our results on loop and suspension functors of augmented algebras in operadic terms.

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