Semi-Free Actions with Manifold Orbit Spaces

Abstract

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 S3-bundles over S2. 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 S3 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.