# On the Theriault conjecture for self homotopy equivalences

### Badr Ben El Krafi

Hassan II University Aïn Chock, Casablanca, Morocco### My Ismail Mamouni

CRMEF Rabat, Morocco

## 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

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.

## Cite this article

Badr Ben El Krafi, My Ismail Mamouni, On the Theriault conjecture for self homotopy equivalences. Rend. Sem. Mat. Univ. Padova 138 (2017), pp. 209–221

DOI 10.4171/RSMUP/138-10