On the Theriault conjecture for self homotopy equivalences (Q1700667)

Let \(\mathrm{aut}_{1}(X)\) be the identity path component of the group of self-homotopy equivalences of a simply connected CW-complex of finite type, \(X\). In this paper, the authors prove that the rational homotopical nilpotency of \(\mathrm{aut}_1(X)\) is less than or equal to the rational cocategory of the classifying space \(\mathrm{Baut}_{1}(X)\). The homotopical nilpotency is due to \textit{I. Berstein} and \textit{T. Ganea} [Ill. J. Math. 5, 99--130 (1961; Zbl 0096.17602)] and the rational cocategory used here was introduced by \textit{M. Sbai} [in: Homotopie algébrique et algèbre locale, Journ. Luminy/France 1982, Astérisque 113--114, 288--291 (1984; Zbl 0548.55003)]. This inequality answers a question of S. Theriault in the particular case of rational spaces.