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The Triviality Conjecture for Spherical Cochains

Author: Wei Yang · Added: September 20, 2026

Article date: September 20, 2026.

Abstract

We prove the triviality conjecture of Heuts and Land: for $n\geq1$ and an $\mathbb{E}_{n+2}$ -ring spectrum $A$ , the augmented cochain algebra $C^*(S^n;A)$ is $\mathbb{E}_{n+1}$ -trivial over $A$ if and only if $A$ is rational. The proof constructs, at each prime $p$ , a $D_p$ -structure on the original multiplication of $A_{(p)}$ . A comparison between configuration spaces and representation spheres turns the operadic obstruction into a functional on cochains. Equivariant normalization and one additional little-disks direction give the required product and unit compatibilities. Hahn’s chromatic-acyclicity theorem and the Hopkins-Smith nilpotence theorem then imply rationality.