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Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action.

Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, and the given circle action on $M$ can be extended to an effective smooth circle action on $N$?

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    $\begingroup$ I don't work in $S^1$-equivariant bordism but there has been some considerable work in this area. Here is a recent paper on the topic. arxiv.org/pdf/1506.04073.pdf $\endgroup$
    – Ryan Budney
    Commented Mar 17 at 6:04
  • $\begingroup$ The quotient of a 3-manifold by a circle action will be an orbifold. If there are no fixed points of the action, then the quotient is a closed orbifold. These are known to be null-cobordant. msp.org/pjm/2000/193-1/p04.xhtml One could then try to extend the circle action to an action over the 3-manifold bounding the orbifold (but I haven’t tried to do this myself). $\endgroup$
    – Ian Agol
    Commented Mar 27 at 18:13

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