The evaluation subgroup of a fibre inclusion

Let xi: X -> j E -> p B be a fibration of simply connected CW complexes of finite type with classifying map h:B -> B aut(1)(X). We study the evaluation subgroup G(n) (E, X; j) of the fibre inclusion as an invariant of the fibre-homotopy type of f. For spherical fibrations, we show the evaluation subgroup may be expressed as an extension of the Gottlieb group of the fibre sphere provided the classifying map h induces the trivial map on homotopy groups. We extend this result after rationalization: We show that the decomposition G(*) (E, X; j) circle times Q = (G(*) (X) circle times Q) circle plus (pi(*) (B) circle times Q) is equivalent to the condition (h(#))(Q) = 0. (c) 2006 Elsevier B.V. All rights reserved.