Lifting obstructions, ordinary obstructions and spherical fibrations

Abstract.

Given a continuous map from a smooth manifold into the base space of a sphere bundle and a lifting, defined outside a regular subset (called the “defect set”), there are obstructions to extend the lifting and there are obstructions to extend the lifting as an ordinary map into the total space. We prove a relation between these two types of obstructions, involving a sort of universal obstruction, and compute this universal obstruction for some concrete examples.