On the Theriault conjecture for self homotopy equivalences. (English) Zbl 1383.55008

Ben El Krafi, Badr; Mamouni, My Ismail

Rend. Semin. Mat. Univ. Padova 138, 209-221 (2017).

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 I. Berstein and T. Ganea [Ill. J. Math. 5, 99–130 (1961; Zbl 0096.17602)] and the rational cocategory used here was introduced by 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.

Reviewer: Daniel Tanré (Villeneuve d’ Ascq)

MSC:

55P62	Rational homotopy theory
55P10	Homotopy equivalences in algebraic topology

Keywords:

Sullivan model; Quillen model; cocategory; homotopical nilpotency; self-homotopy equivalences

Citations:

Zbl 0096.17602; Zbl 0548.55003