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Let $M$ be a closed, orientable 3-manifold with a non-trivial differentiable $S^1$-action.

What does this imply for $M$? What are examples except for (products of) spheres?

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The standard examples are Seifert fibre spaces, which admit a free $S^1$ action. The wikipedia page gives a reasonable description and some references. – HJRW Oct 10 '11 at 18:26
@HW: they are standard examples, but not the only examples. In particular, if you allow fixed points, you get direct sums of lens spaces (proved in the Raymond paper, where a complete classification is given). – Igor Rivin Oct 10 '11 at 19:32
up vote 10 down vote accepted

You will find the complete answer in the papers

Frank Raymond, Classification of the actions of the circle on 3-manifolds. Trans. Amer. Math. Soc. 131 1968 51–78.


Peter Orlik and Frank Raymond, Actions of SO(2) on 3-manifolds. 1968 Proc. Conf. on Transformation Groups (New Orleans, La., 1967) pp. 297–318 Springer, New York

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He will find the complete answer only if he finds the second paper, which is nontrivial. – Igor Rivin Oct 10 '11 at 18:30
Only nontrivial if you don't know where to look :) – José Figueroa-O'Farrill Oct 10 '11 at 19:19
Is interlibrary loan really nontrivial these days? – Richard Kent Oct 10 '11 at 19:39
You don't know the OP's situation. He might be thinking these things in the jungles of amazon (though his web page indicates that he is in Bonn, and the book is, with high probability, in the MPI library, I admit). – Igor Rivin Oct 10 '11 at 19:42
You've got me there. – Richard Kent Oct 10 '11 at 20:12

If the action has no fixed points, that class of manifolds has a name: Seifert-fibred 3-manifold with an orientation on the fiber. Seifert classified a slightly larger class of manifolds (Seifert-fibered ones -- no orientability constraint on the fibers). This is available in either the Jaco or Hatcher 3-manifolds lecture notes, also Orlik's book on Seifert-Fibered spaces.

Such examples include things like lens spaces, and relatively complicated 3-manifolds like the double of a torus knot complement.

When you allow fixed points, the orbit decomposition theorem gives you a stratification of the manifold into the fixed-point set (a link) together with its Seifert-fibred complement. So you could view these manifolds as certain Dehn Fillings on Seifert-fibred manifolds with boundary. As the references above point out, they're abundant.

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