# A gluing formula for families Seiberg-Witten invariants

## TL;DR

This paper establishes a gluing formula for families Seiberg-Witten invariants of 4-manifold families formed by fiberwise connected sum, linking these invariants to those of individual summands and exploring various applications.

## Contribution

It introduces a new gluing formula for families Seiberg-Witten invariants, including variants with twists and equivariance, expanding tools for 4-manifold topology.

## Key findings

- Derived a gluing formula relating family invariants to summand invariants
- Constructed variants incorporating charge conjugation and group actions
- Applied the formula to obstructions in smooth topology and scalar curvature

## 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.