A surgery formula for Seiberg-Witten invariants

We prove a surgery formula for the ordinary Seiberg-Witten invariants of smooth $4$-manifolds with $b_1 =1$. Our formula expresses the Seiberg-Witten invariants of the manifold after the surgery, in terms of the original Seiberg-Witten moduli space cut down by a cohomology class in the configuration space. This formula can be used to find exotic smooth structures on nonsimply connected $4$-manifolds, and gives a lower bound of the genus of an embedding surface in nonsimply connected $4$-manifolds. In forthcoming work, we will extend these results to give a surgery formula for the families Seiberg-Witten invariants.