Non-Formality of $S^2$ via the free loop space

TL;DR

This paper demonstrates that the algebraic structures associated with the 2-sphere's cochains are not formal, revealing a discrepancy in the expected algebraic-topological correspondence in string topology.

Contribution

It explicitly computes the defect in the $E_ullet$-structure of the cochains of $S^2$ and relates it to recent developments in string topology.

Findings

The $E_1$-equivalence does not preserve the inclusion of constant loops.

The explicit defect is computed in terms of the $E_$-structure.

The results connect to recent work by Poirier-Tradler on string topology.

Abstract

We show that the $E_1$-equivalence $C^\bullet(S^2) \simeq H^\bullet(S^2)$ does not intertwine the inclusion of constant loops into the free loop space $S^2 \to LS^2$. That is, the isomorphism $HH_\bullet(H^\bullet(S^2)) \cong H^\bullet(LS^2)$ does not preserve the obvious maps to $H^\bullet(S^2)$ that exist on both sides. We give an explicit computation of the defect in terms of the $E_\infty$-structure on $C^\bullet(S^2)$. Finally, we relate our calculation to recent work of Poirier-Tradler on the string topology of $S^2$.