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.