Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory

Abstract

Let $X$ be a smooth, compact, oriented $4$-manifold. Building upon work of Li-Liu, Ruberman, Nakamura and Konno, we consider a families version of Seiberg-Witten theory and obtain obstructions to the existence of certain group actions on $X$ by diffeomorphisms. The obstructions show that certain group actions on $H^2(X , \mathbb{Z})$ preserving the intersection form can not be lifted an action of the same group on $X$ by diffeomorphisms. Using our obstructions, we construct numerous examples of group actions which can be realised continuously but can not be realised smoothly for any differentiable structure. For example, we construct compact simply-connected $4$-manifolds $X$ and involutions $f : H^2(X , \mathbb{Z}) \to H^2(X , \mathbb{Z})$ such that $f$ can be realised by a continuous involution on $X$ or by a diffeomorphism, but not by an involutive diffeomorphism for any smooth structure on $X$.