On fibre spaces and nilpotency. II

Extract

1. Introduction. Let X be a space and let E be a fibre space over E. A fibre-preserving map f: E → E determines, for each point x ∈ X, a map f_x: E_x → E_x of the fibre over x. In a previous note (3) the situation was considered where f_x is null-homotopic, for all x. In the present note we turn our attention to the situation where f_x is homotopic to the identity on E_x, for all x ∈ X. If X admits a numerable categorical covering (as when X is an ANR) then such a fibre-preserving map f is a fibre homotopy equivalence, by the well-known theorem of Dold(1). Then the set Φ₁(E) of fibre homotopy classes of such maps forms a normal subgroup of the group Φ_*(E) of fibre homotopy classes of fibre homotopy equivalences. The purpose of this note is to prove

Theorem 1.1. Let X be a paracompact space of finite category. Let E be a fibre bundle over X of which the fibres are compact and path-connected ANR's. Then the Φ_*(E)-growp Φ₁(E) is Φ_*(E)-nilpotent of class ≤ cat X.