On the Theriault conjecture for self homotopy equivalences

Abstract

Our main purpose in this paper is to resolve, in a rational homotopy theory context, the following open question asked by S. Theriaul: given a topological space $X$, what one may say about the nilpotency of aut$\_1(X)$ when the cocategory of its classifying space Baut$\_1(X)$ is finite? Here aut$\_1(X)$ denotes the path component of the identity map in the set of self homotopy equivalences of $X$. More precisely, we prove that $$ \mathrm {HNil}\_\mathbb Q(\mathrm {aut}1(X))\leqslant\mathrm {cocat}\mathbb Q(\mathrm {Baut}\_1(X)), $$ when $X$ is a simply connected CW-complex of finite type and that the equality holds when Baut$\_1(X)$ is coformal. Many intersections with other popular open questions will be discussed.