On the orders of periodic diffeomorphisms of 4-manifolds

TL;DR

This paper explores the limitations on the order of prime cyclic group actions on smooth 4-manifolds that do not admit circle actions, revealing topological and smooth-structure-dependent bounds.

Contribution

It provides affirmative results for holomorphic and symplectic actions, establishing bounds on prime order actions based on the manifold's structure.

Findings

Bound C is topological for holomorphic actions.

Bound C depends on smooth structure for symplectic actions.

No large prime order actions exist on certain 4-manifolds without circle actions.

Abstract

This paper initiated an investigation on the following question: Suppose a smooth 4-manifold does not admit any smooth circle actions. Does there exist a constant $C>0$ such that the manifold support no smooth $\Z_p$-actions of prime order for $p>C$? We gave affirmative results to this question for the case of holomorphic and symplectic actions, with an interesting finding that the constant $C$ in the holomorphic case is topological in nature while in the symplectic case it involves also the smooth structure of the manifold.