The evaluation subgroup of a fibre inclusion

Let ξ : 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 ξ. 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 ) ⊗ Q = ( G ∗ ( X ) ⊗ Q ) ⊕ ( π ∗ ( B ) ⊗ Q ) is equivalent to the condition ( h ♯ ) Q = 0 .