Bauer–Furuta invariants under $${\mathbb{Z}_2}$$ -actions

Abstract

S. Bauer and M. Furuta defined a stable cohomotopy refinement of the Seiberg–Witten invariants. In this paper, we prove a vanishing theorem of Bauer–Furuta invariants for 4-manifolds with smooth \({\mathbb{Z}_2}\) -actions. As an application, we give a constraint on smooth \({\mathbb{Z}_2}\) -actions on homotopy K3#K3, and construct a nonsmoothable locally linear \({\mathbb{Z}_2}\) -action on K3#K3. We also construct a nonsmoothable locally linear \({\mathbb{Z}_2}\) -action on K3.