A gluing formula for families Seiberg–Witten invariants

Abstract

We prove a gluing formula for the families Seiberg–Witten invariants of families of $4$–manifolds obtained by fibrewise connected sum. Our formula expresses the families Seiberg–Witten invariants of such a connected sum family in terms of the ordinary Seiberg–Witten invariants of one of the summands, under certain assumptions on the families. We construct some variants of the families Seiberg–Witten invariants and prove the gluing formula also for these variants. One variant incorporates a twist of the families moduli space using the charge conjugation symmetry of the Seiberg–Witten equations. The other variant is an equivariant Seiberg–Witten invariant of smooth group actions. We consider several applications of the gluing formula, including obstructions to smooth isotopy of diffeomorphisms, computation of the mod $2$ Seiberg–Witten invariants of spin structures, and relations between mod $2$ Seiberg–Witten invariants of $4$–manifolds and obstructions to the existence of invariant metrics of positive scalar curvature for smooth group actions on $4$–manifolds.