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Hurwitz-type bound, knot surgery, and smooth \({\mathbb{S }}^1\)-four-manifolds

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In this paper we prove several related results concerning smooth \(\mathbb{Z }_p\) or \(\mathbb{S }^1\) actions on \(4\)-manifolds. We show that there exists an infinite sequence of smooth \(4\)-manifolds \(X_n\), \(n\ge 2\), which have the same integral homology and intersection form and the same Seiberg-Witten invariant, such that each \(X_n\) supports no smooth \(\mathbb{S }^1\)-actions but admits a smooth \(\mathbb{Z }_n\)-action. In order to construct such manifolds, we devise a method for annihilating smooth \(\mathbb{S }^1\)-actions on \(4\)-manifolds using Fintushel-Stern knot surgery, and apply it to the Kodaira-Thurston manifold in an equivariant setting. Finally, the method for annihilating smooth \(\mathbb{S }^1\)-actions relies on a new obstruction we derived in this paper for existence of smooth \(\mathbb{S }^1\)-actions on a \(4\)-manifold: the fundamental group of a smooth \(\mathbb{S }^1\)-four-manifold with nonzero Seiberg-Witten invariant must have infinite center. We also include a discussion on various analogous or related results in the literature, including locally linear actions or smooth actions in dimensions other than four.

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Acknowledgments

I wish to thank Ron Fintushel for communications regarding the knot surgery formula for Seiberg-Witten invariants, Dieter Kotschick for bringing to my attention his related work, and Inanc Baykur for helpful comments. Part of this work was done during the author’s visits to the Max Planck Institute for Mathematics, Bonn and the Institute for Advanced Study, Princeton. I wish to thank both institutes for their hospitality and financial support. Finally, I wish to thank Slawomir Kwasik for his timely advices, particularly his help with the related results in the literature.

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  1. University of Massachusetts, Amherst, MA, USA

    Weimin Chen

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  1. Weimin Chen
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Correspondence to Weimin Chen.

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The author is partially supported by NSF Grant DMS-1065784.

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Chen, W. Hurwitz-type bound, knot surgery, and smooth \({\mathbb{S }}^1\)-four-manifolds. Math. Z. 276, 267–279 (2014). https://doi.org/10.1007/s00209-013-1198-x

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