  We construct a simple topological invariant of certain 3-manifolds, including
quotients of the 3-sphere by finite groups, based on the fact that the tangent
bundle of an orientable 3-manifold is trivialisable. This invariant is strong
enough to yield the classification of lens spaces of odd, prime order. We also
use properties of this invariant to show that there is an oriented 3-manifold
with no universally tight contact structure. We generalise and sharpen this
invariant to an invariant of a finite covering of a 3-manifold.
