2013/03/04 by Weimin Chen, Chen, Weimin
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1303.0848
openalex publication_date 2013/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we prove several related results concerning smooth \Zp or \s1 actions on 4-manifolds. We show that there exists an infinite sequence of smooth 4-manifolds Xn, n≥ 2, which have the same integral homology and intersection form and the same Seiberg-Witten invariant, such that each Xn supports no smooth \s1-actions but admits a smooth \Zn-action. In order to construct such manifolds, we devise a method for annihilating smooth \s1-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 \s1-actions relies on a new obstruction we derived in this paper for existence of smooth \s1-actions on a 4-manifold: the fundamental group of a smooth \s1-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.