Hurwitz-type bound, knot surgery, and smooth $\s^1$-four-manifolds

TL;DR

This paper constructs infinite families of 4-manifolds with identical homology and invariants that support no smooth circle actions but admit finite cyclic actions, using knot surgery and a new fundamental group obstruction.

Contribution

It introduces a method to eliminate smooth $ ext{s}^1$-actions on 4-manifolds via knot surgery and establishes a new obstruction based on the fundamental group's center.

Findings

Existence of infinite 4-manifold families with identical invariants but differing circle actions.

A new obstruction: nonzero Seiberg-Witten invariant implies infinite center in the fundamental group.

Application of knot surgery to control smooth actions on 4-manifolds.

Abstract

In this paper we prove several related results concerning smooth $\Z_p$ or $\s^1$ actions on 4-manifolds. We show that there exists an infinite sequence of smooth 4-manifolds $X_n$, $n\geq 2$, which have the same integral homology and intersection form and the same Seiberg-Witten invariant, such that each $X_n$ supports no smooth $\s^1$-actions but admits a smooth $\Z_n$-action. In order to construct such manifolds, we devise a method for annihilating smooth $\s^1$-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 $\s^1$-actions relies on a new obstruction we derived in this paper for existence of smooth $\s^1$-actions on a 4-manifold: the fundamental group of a smooth $\s^1$-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.