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

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1997

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Huck, Wolfgang

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Zusammenfassung

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.



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.



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.

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ISO 690HUCK, Wolfgang, 1997. A note on circle actions on 5- and 6-dimensional manifolds
BibTex
@unpublished{Huck1997circl-6188,
  year={1997},
  title={A note on circle actions on 5- and 6-dimensional manifolds},
  author={Huck, Wolfgang}
}
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    <dcterms:abstract xml:lang="eng">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.&lt;br /&gt;&lt;br /&gt;&lt;br /&gt;&lt;br /&gt;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.&lt;br /&gt;&lt;br /&gt;&lt;br /&gt;&lt;br /&gt;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.</dcterms:abstract>
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