Abstract
Let Z denote an irreducible homology 3-sphere. The actions of SO(2) on Σ are given in [2]. The involution imbedded in such an action is free if and only if all the α j are odd. In [1] Medrano constructs examples of free involutions on certain Σ, but it is not known whether these are imbedded in SO(2) actions. More generally:
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Q: Is every free involution of an irreducible homology 3-sphere imbedded in some SO(2) action?
By Livesay, the answer is yes for S3. The examples below show that the conditions of Q are necessary.
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References
Lopez de Medrano, S.: Involutions of homotopy spheres and homology 3-spheres, Bull. Amer. Math. Soc. 73, 727–731 (1967).
Orlik, P., and F. Raymond: Actions of SO(2) on 3-manifolds, these Proceedings.
Orlik, P., On 3-manifolds with local SO(2) action to appear in Quar. J. Math.Oxford).
Stallings, J.: On fibering certain 3-manifolds, Topology of 3-manifolds, Prentice Hall 95–100 (1962).
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© 1968 Springer-Verlag Berlin · Heidelberg
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Orlik, P. (1968). Examples of Free Involutions on 3-Manifolds. In: Mostert, P.S. (eds) Proceedings of the Conference on Transformation Groups. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46141-5_11
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DOI: https://doi.org/10.1007/978-3-642-46141-5_11
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