We prove a gluing formula for the families Seiberg–Witten invariants of families of
–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
Seiberg–Witten invariants of spin structures, and relations between mod
Seiberg–Witten
invariants of
–manifolds
and obstructions to the existence of invariant metrics of positive scalar curvature for smooth group
actions on
–manifolds.
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