# Semi-Free Actions with Manifold Orbit Spaces

### John Harvey

Department of Mathematics, Swansea University, Swansea SA1 8EN, United Kingdom### Martin Kerin

School of Mathematics, Statistics \& Applied Mathematics, NUI Galway, Ireland### Krishnan Shankar

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

## 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 $\mathbf{S}^3$-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.

## Cite this article

John Harvey, Martin Kerin, Krishnan Shankar, Semi-Free Actions with Manifold Orbit Spaces. Doc. Math. 25 (2020), pp. 2085–2114

DOI 10.4171/DM/794