A bound for the nilpotency of a group of self homotopy equivalences

Abstract

Let E Ω ( X ) \mathcal {E}_\Omega (X) be the group of homotopy classes of self-homotopy equivalences of X X such that Ω f ≃ 1 d Ω X \Omega f\simeq 1d_{\Omega X} . We prove that E Ω ( X ) \mathcal {E}_\Omega (X) is a nilpotent group and that nil ⁡ E Ω ( X ) ≤ cat ⁡ ( X ) − 1 \operatorname {nil} \mathcal {E}_{\Omega }(X)\le \operatorname {cat}(X)-1 .