  In this paper we describe a method to establish when a symplectic manifold
$M$ with semi-free Hamiltonian $S^{1}$-action is unique up to isomorphism
(equivariant symplectomorphism). This will rely on a study of the symplectic
topology of the reduced spaces. We prove that if the reduced spaces satisfy a
rigidity condition, then the manifold $M$ is uniquely determined by fixed point
data. In particular we can prove that there is a unique family up to
isomorphism of 6-dimensional symplectic manifolds with semi-free $S^1$-action
and isolated fixed points.
