A note on circle actions on 5- and 6-dimensional manifolds

Abstract

M. Aubry and J.-M. Lemaire proved in 1985 that a simply connected 5-manifold M admits a free circle action if and only if H*(M) is free or, what amounts to the same, if and only if M is a connected sum of S3-bundles over S2. We show that H*(M) being free is as well equivalent to the existence of a semifree action (or, even better: to the existence of an action with only fixed points, free and isolated exceptional orbits) on M.<br /><br /><br /><br />Moreover, we use results on Brieskorn manifolds due to J. Milnor to prove that every irreducible (i.e not splittable into a non-trivial connected sum) simply connected spin-5-manifold can be endowed with a fixed point free circle action.<br /><br /><br /><br />Finally we show that a differentiable, simply connected 6-manifold with free cohomology admits a free S1-action if and only if it admits an action without fixed points.