Semi-free actions with manifold orbit spaces (Q2213800)

A smooth action of a Lie group \(G\) on a manifold \(M\) is called semi-free if there are only free orbits and fixed points. In general the orbit space of such an action is not a manifold. However it is a manifold if \(G=S^1\) and the codimension of the fixed set is four or \(G=S^3\) and the codimension of the fixed set is eight. In the paper under review simply connected closed \(5\)-manifolds with semi-free circle actions and one-dimensional fixed sets are classified up to equivariant and non-equivariant diffeomorphism. The proof of the non-equivariant classification uses the Barden-Smale classification of simply connected \(5\)-manifolds with torsion free second homology group. The equivariant classification is a simplification of a more general result due to Levine in the special case at hand. Moreover, simply connected closed \(8\)-manifolds with semi-free \(S^3\)-action and isolated fixed points are studied.