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 diffeomorpihsms, computation of the mod $2$ Seiberg-Witten invariants of spin structures, 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.

arXiv comment54 pages

References in corpus (2)

• Positive scalar curvature, diffeomorphisms and the Seiberg-Witten invariants
• Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory

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• Positive scalar curvature and higher-dimensional families of Seiberg-Witten equations
• Non-trivial smooth families of $K3$ surfaces
• The homology of moduli spaces of 4-manifolds may be infinitely generated