On the relation between lifting obstructions and ordinary obstructions

We consider partial liftings k :A→E of maps f :X→B, where (X,A) is a relative CW-complex and E→B is a fibration. In this situation, we have a primary obstruction to extend the partial lifting to a lifting of f on all of X, and there is an obstruction to extend k as an ordinary map into the space E. A relation between these two cohomology classes is proved when the fibre of E→B is an Eilenberg–MacLane space K(Π,n) and πi(E)=0 for i≤q−1, where q≥n+2, that specializes to well-known formulas about secondary obstructions. The result is applied to the Hopf fibration (what includes defect sections in S1-bundles over 4-manifolds) and to the case of a certain SU(3)-bundle over S4.