Gluing formula for the stable cohomotopy version of Seiberg-Witten invariants along 3-manifolds with b₁ > 0

We will define a version of Seiberg-Witten-Floer stable homotopy types for a closed, oriented 3-manifold Y with b₁(Y) > 0 and a spin-c structure c on Y with c₁(c) torsion under an assumption on Y. Using the Seiberg-Witten-Floer stable homotopy type, we will construct a gluing formula for the stable cohomotopy version of Seiberg-Witten invariants of a closed 4-manifold X which has a decomposition X = X₁ ∪Y X₂ along Y.