  In this paper, we study smooth, semi-free actions on closed, smooth, simply
connected manifolds, such that the orbit space is a smoothable manifold. We
show that the only simply connected $5$-manifolds admitting a smooth, semi-free
circle action with fixed-point components of codimension $4$ are connected sums
of $S^3$-bundles over $S^2$. Furthermore, the Betti numbers of the
$5$-manifolds and of the quotient $4$-manifolds are related by a simple formula
involving the number of fixed-point components. We also investigate semi-free
$S^3$ actions on simply connected $8$-manifolds with quotient a $5$-manifold
and show, in particular, that there are strong restrictions on the topology of
the $8$-manifold.
