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 \(\mathbf{S}^3\)-bundles over \(\mathbf{S}^2\). 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.

57S17, 55R55, 57K50

circle action, semi-free action, \(5\)-manifolds, \(4\)-manifolds, \(8\)-manifolds

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Department of Mathematics, Swansea University, Swansea SA1 8EN, United Kingdom

School of Mathematics, Statistics \& Applied Mathematics, NUI Galway, Ireland

Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA