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

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 $S^2\to L{S}^2$. That is, the isomorphism $H{H}_{\bullet }(H^{\bullet }(S^2))\cong H^{\bullet }(L{S}^2)$ 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 $S^2$.

arXiv comment12 pages