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

TL;DR

This paper investigates the formality properties of $ ext{E}_n$-algebras, especially the cochain algebra of spheres, revealing its formality over various coefficients and connecting operadic structures to algebraic properties.

Contribution

It proves the formality of the cochain algebra of spheres as an $ ext{E}_n$-algebra over general coefficients and establishes a fully faithful functor from operads to monads in spectra.

Findings

The cochain algebra of the $n$-sphere is formal as an $ ext{E}_n$-algebra.

Formality holds over general commutative ring spectra, but not $ ext{E}_{n+1}$-formal unless coefficients are rational.

The free functor from operads to monads in spectra is fully faithful on a significant subcategory.

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.