Abstract
In this thesis, We first 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 non-simply connected $4$-manifolds, and gives a lower bound of the genus of an embedding surface in nonsimply connected $4$-manifolds.
In the second part, We generalize the $S^1$-equivariant family Bauer-Furuta invariant to non-simply connected manifolds, and construct a refinement of this invariant. We use it to show that if $X_1,X_2$ are two homology tori whose determinants $r_1,r_2$ are odd, then the Dehn twist along a $3$-sphere in the neck of $X_1\mathbin{\#} X_2$ is not smoothly isotopic to the identity.