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

TL;DR

This paper develops a families Seiberg-Witten theory to identify obstructions to smooth group actions on 4-manifolds, showing certain topological actions cannot be realized smoothly, thus revealing new smooth structure invariants.

Contribution

It introduces a families Seiberg-Witten framework to detect obstructions to lifting topological group actions to smooth diffeomorphisms on 4-manifolds.

Findings

Constructed examples of topological actions not realizable smoothly

Identified obstructions to lifting group actions via Seiberg-Witten invariants

Demonstrated differences between continuous and smooth realizations of involutions

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 $X$.