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
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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 55-manifolds admitting a smooth, semi-free circle action with fixed-point components of codimension 44 are connected sums of S3\mathbf{S}^3-bundles over S2\mathbf{S}^2. Furthermore, the Betti numbers of the 55-manifolds and of the quotient 44-manifolds are related by a simple formula involving the number of fixed-point components. We also investigate semi-free S3S^3 actions on simply connected 88-manifolds with quotient a 55-manifold and show, in particular, that there are strong restrictions on the topology of the 88-manifold.

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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