A general connected sum formula for the families Bauer-Furuta invariant

The Bauer-Furuta invariant of a family of smooth 4-manifolds is a stable cohomotopy refinement of the families Seiberg-Witten invariant and is constructed from a finite dimensional approximation of the Seiberg-Witten monopole map. We prove a general formula for the families Bauer-Furuta invariant of a fibrewise connected sum, extending Bauer’s non-parameterised formula. In a subsequent paper, we will use this formula to derive a general connected sum formula for the families Seiberg-Witten invariant which incorporates both the families blow-up formula of Liu and the gluing formula of Baraglia-Konno.