Vector bundles over three-dimensional spherical space forms

In this work we consider the class of the compact connected three-dimensional manifolds with positive constant curvature, also known as the three-dimensional spherical space forms. These spaces, or subclasses like generalized quaternions or lens spaces, appear in many different contexts in topology and geometry, and have been completely classified; it is thus natural to ask if we can also count the bundles over them. We answer positively to this question, and give tables in Section 5 to describe all the vector bundles of rank less than 3 over any three-dimensional spherical space form. Besides, in Section 2, we show that, under reasonably wide assumptions on the structure group G, G-bundles over any low (lower or equal to three)-dimensional manifolds can be counted effectively.