Nilpotent subgroups of the group of fibre homotopy equivalences

Let ξ = (E, p,B, F) be a Hurewicz fibration. In this paper we study the space LG(ξ) consisting of fibre homotopy self equiva-lences of ξ inducing by restriction to the fibre a self homotopy equivalence of F belonging to the group G. We give in particular conditions implying that π1(LG(ξ)) is finitely generated or that L1(ξ) has the same rational homotopy type as aut1(F). Let ξ = (E, p,B, F) be a Hurewicz fibration where B and F are com-pactly generated spaces. The set of free (not necessarily fibre) homotopy classes of free fibre homotopy equivalences of ξ into itself is a group L(ξ), for the multiplication induced by the composition of maps. Recall that a fibre homotopy equivalence f: E → E induces an ho-motopy equivalence of p−1(b) for each b ∈ B (A theorem of Dold ([4, Theorem 6.3]) asserts that the converse is true if B is a CW complex). There exists thus a natural ma