Pullbacks of Sphere Fibrations over Connected Sums

We prove conditions under which the total space of the pullback of a sphere fibration over a connected sum is homotopy equivalent to a connected sum with a gyration. Existing results of this type often depend on geometric methods. We develop new methods based only on homotopy theory, allowing for generalisations from manifolds to Poincar\'e Duality complexes and from integral settings to local ones. Several applications are given.