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Question: Suppose you have a simply connected, closed, orientable, smooth manifold $M$. What are some restrictions to the existence of a smooth (non-trivial) $S^{1}$-action on $M$?

Note: 1.There are some well known restrictions in the case that we require this action to have isolated fixed points (or preserve some additional structures), but I am not aware of any obstruction in this generality.

  1. I am aware of the classification of manifolds with $S^{1}$-actions in dimension $2$ and $3$. In higher dimensions (say $\geq 6$) it seems to become less clear.

  2. I am particularly interested in dimension 6. I would like to get a picture of which manifolds in this dimension (satisying the above restrictions), have a smooth $S^{1}$-action.

Edit: Out of curiosity, using Puppe's methods, is it possible to construct a simply connected, smooth, complex projective 3-fold such that the underlying smooth manifold has no circle actions?

Edit 2 It turns out that (the underlying smooth 6-manifold of) any smooth hypersurface in $\mathbb{C}\mathbb{P}^4$ of degree atleast 3 has no smooth (non-trivial) circle actions. It follows from the main result of https://arxiv.org/pdf/1108.5327.pdf.

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  • $\begingroup$ What is your definition of "nontrivial"? $\endgroup$
    – Igor Rivin
    Commented Feb 18, 2018 at 19:52
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    $\begingroup$ I am not sure about dimension $6$ but certainly there are obstructions in dimensions $4k$, namely, the $\hat A$-genus of a closed spin manifold with a smooth circle action is zero (by a famous result of M. Atiyah and F.Hirzebruch). Also to add to your low dimensional results look at R. Fintushel's "Classification of circle actions on 4-manifolds", Trans. Amer. Math. Soc. 242 (1978), 377–390. $\endgroup$
    – Igor Belegradek
    Commented Feb 18, 2018 at 20:01
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    $\begingroup$ I suspect you already know this since you focus on 6-manifolds but in any case there are published papers of V.Puppe ("Simply connected manifolds without $S^1$-symmetry" and "Do manifolds have little symmetry?" ) and M.Kreck ("Simply connected asymmetric manifolds") that claim obstructions to nontrivial smooth $S^1$-actions. Kreck's result has been retracted. I am not sure about the exact status of Puppe's work but from the mathscinet review to Kreck's retraction I gather the problem is still open. $\endgroup$
    – Igor Belegradek
    Commented Feb 18, 2018 at 20:40
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    $\begingroup$ In particular, Puppe's "Simply connected 6-dimensional manifolds with little symmetry and algebras with small tangent space" [Ann. of Math. Stud., 1994] claims to produce a simply-connected closed 6-manifold with no orientation-preserving actions of finite groups of prime order. The difficulty seems to be to rule out the orientation reversing actions. But any circle action is orientation preserving (because the circle is connected). $\endgroup$
    – Igor Belegradek
    Commented Feb 18, 2018 at 20:48
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    $\begingroup$ Thanks to Stolz's solution to the Gromov-Lawson conjecture, it follows from the result mentioned in Igor Belegradek's first comment that a simply connected spin manifold of dimension $4k > 4$ with an $S^1$-action admits a metric of positive scalar curvature. $\endgroup$
    – Michael Albanese
    Commented Feb 18, 2018 at 20:49

2 Answers 2

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V. Puppe, in Simply connected manifolds without $S^1$-symmetry. Algebraic topology and transformation groups, Proc. Conf., Göttingen/FRG 1987, Lect. Notes Math. 1361, 261-268 (1988) proved the following:

There exist a simply connected, closed, oriented smooth 6-dimensional manifold $M$ such that no closed, orientable manifold with the same rational cohomology algebra as $M$ admits a non-trivial circle action.

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It is a result of Atiyah-Hirzebruch (1970) that the $\hat{A}$ genus of a spin manifold with a nontrivial $S^1$ action vanishes, and a result of Herrera and Herrera that the same result, if the manifold is not necessarily spin, but has finite $\pi_2,$ then the same result, is true. In the meantime, there is the result of Freedman and Meeks (Une obstruction élémentaire à l’existence d’une action continue de groupe dans une variété, C. R. Acad. Sci., Paris, Sér. A 286, 195-198 (1978)) that there are some cohomological/geometric obstructions (so the connected sum of a non-sphere and a torus admits no circle action) - this has been generalized in

Assadi, Amir; Burghelea, Dan, Examples of asymmetric differentiable manifolds, Math. Ann. 255, 423-430 (1981). ZBL0437.57021..

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    $\begingroup$ The example of a torus and a non-sphere is irrelevant, since I required explicitly that the manifold is simply connected. $\endgroup$
    – Nick L
    Commented Feb 18, 2018 at 20:38
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    $\begingroup$ @NickL Please don't be rude. The example I alluded to is in the paper, doubtlessly, the same technique (or its generalizations) can be used to construct simply connected examples. $\endgroup$
    – Igor Rivin
    Commented Feb 18, 2018 at 21:55
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    $\begingroup$ @IgorRivin: the simply-connected case is really quite different. To date there is not a single example of a closed simply-connected smooth manifold that admit no smooth action by a compact Lie group. By contrast, many non-simply-connected examples are known. $\endgroup$
    – Igor Belegradek
    Commented Feb 18, 2018 at 22:06
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    $\begingroup$ @ Igor Rivin Sorry, my previous comment was a bit blunt. Good to know, thanks! $\endgroup$
    – Nick L
    Commented Feb 18, 2018 at 22:07
  • $\begingroup$ @IgorBelegradek Interesting, thanks! $\endgroup$
    – Igor Rivin
    Commented Feb 18, 2018 at 22:09

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