  Let the circle act effectively in a Hamiltonian fashion on a compact
symplectic manifold $(M, \omega)$. Assume that the fixed point set $M^{S^1}$
has exactly two components, $X$ and $Y$, and that $\dim(X) + \dim(Y) +2 =
\dim(M)$. We first show that $X$, $Y$ and $M$ are simply connected. Then we
show that, up to $S^1$-equivariant diffeomorphism, there are finitely many such
manifolds in each dimension. Moreover, we show that in low dimensions, the
manifold is unique in a certain category. We use techniques from both areas of
symplectic geometry and geometric topology.
