On the Theriault conjecture for self homotopy equivalences

Ben El Krafi, Badr; Mamouni, My Ismail

Rendiconti del Seminario Matematico della Università di Padova, Tome 138 (2017), pp. 209-221.

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${}_1\left(X\right)$ when the cocategory of its classifying space Baut${}_1\left(X\right)$ is finite? Here aut${}_1\left(X\right)$ denotes the path component of the identity map in the set of self homotopy equivalences of $X$. More precisely, we prove that

$${\mathrm{HNil}}_{\mathbb{Q}}\left({\mathrm{aut}}_1\left(X\right)\right)⩽{\mathrm{cocat}}_{\mathbb{Q}}\left({\mathrm{Baut}}_1\left(X\right)\right),$$

when $X$ is a simply connected CW-complex of finite type and that the equality holds when Baut${}_1\left(X\right)$ is coformal. Many intersections with other popular open questions will be discussed.

Classification : 55
Mots-clés : Rational homotopy theory, Sullivan minimal models, Quillen minimal models, self homotopy equivalences, classifying space, cocategory, homotopical nilpotency, Lie derivation