Bonn Topology Group - Abstracts

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Talk

June 25, 2024
Gijs Heuts (Universiteit Utrecht): Formality of E_n-algebras and cochains on spheres

Abstract

It is a classical fact of rational homotopy theory that the E_infinity-algebra of rational cochains on a sphere is formal, i.e., quasi-isomorphic to the cohomology of the sphere. In other words, this algebra is square-zero. This statement fails with integer or mod p coefficients. We show, however, that the cochains of the n-sphere are still E_n-trivial with coefficients in arbitrary cohomology theories. This is a consequence of a more general statement on (iterated) loops and suspensions of E_n-algebras, closely related to Koszul duality for the E_n-operads. We will also see that these results are essentially sharp: if the R-valued cochains of S^n have square-zero E_{n+1}-structure (for some rather general ring spectrum R), then R must be rational. This is joint work with Markus Land.