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 → Baut1(X). We study the evaluation subgroup Gn(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.