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


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 XX, what one may say about the nilpotency of aut1(X)_1(X) when the cocategory of its classifying space Baut1(X)_1(X) is finite? Here aut1(X)_1(X) denotes the path component of the identity map in the set of self homotopy equivalences of XX. More precisely, we prove that

HNilQ(aut1(X))cocatQ(Baut1(X)),\mathrm {HNil}_\mathbb Q(\mathrm {aut}_1(X))\leqslant\mathrm {cocat}_\mathbb Q(\mathrm {Baut}_1(X)),

when XX is a simply connected CW-complex of finite type and that the equality holds when Baut1(X)_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