A generalized Mayer-Vietoris sequence involving crossed homomorphisms is established and the construction is applied to the homotopy sequence of the CW-pair (X.X1) to relate the homotopy sequences of (X.X1) and the fibre bundle F → E → X in low dimensions. If there is a partial cross-section of E → X over X2, the classical form, π1E ∼ π1[xtilde] π1F as a semidirect product, results. In case there is no extension over X2of any cross-section of the restricted bundle χ:π2(x2, x1) → X1the corresponding obstruction map XE:π2(x2,x1) → π1F is non-trivial and in case F → E → X is an SO(n)-bundle (n ≥ 3), χEmaps into a subgroup of the centre, Z(π1F), of order at most 2.