Relative (non-)formality of the little cubes operads and the algebraic Cerf Lemma

Abstract

It is shown that the operad maps $E_n\to E_{n+k}$ are formal over the reals for $k\geq 2$ and non-formal for $k=1$. Furthermore we compute the cohomology of the deformation complex of the operad maps $E_{n}\to E_{n+1}$, proving an algebraic version of the Cerf Lemma.