## Semi-Free Actions with Manifold Orbit Spaces

##### Doc. Math. 25, 2085-2114 (2020)
DOI: 10.25537/dm.2020v25.2085-2114

### Summary

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.

### Mathematics Subject Classification

57S17, 55R55, 57K50

### Keywords/Phrases

circle action, semi-free action, $5$-manifolds, $4$-manifolds, $8$-manifolds

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### Affiliation

Harvey, John
Department of Mathematics, Swansea University, Swansea SA1 8EN, United Kingdom
Kerin, Martin
School of Mathematics, Statistics \& Applied Mathematics, NUI Galway, Ireland
Shankar, Krishnan
Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA