The Seiberg-Witten equations and four-manifolds with boundary

An early result by Donaldson says that if Z is closed and JZ is negative definite then JZ is isomorphic to some diagonal form 〈−1〉 ⊕ · · · ⊕ 〈−1〉. More generally one may ask which negative definite forms can occur if Z is allowed to have some fixed oriented rational homology sphere Y as boundary. The main purpose of the present paper is to apply the equations recently introduced by Seiberg and Witten W to prove a finiteness result about the definite forms associated to an arbitrary Y . It is useful to consider the more general situation where the boundary of Z is a disjoint union of rational homology spheres: ∂Z = Y1 ∪ · · · ∪ Yl. (Of course, ∪jYj and #jYj bound the same intersection forms, since the standard cobordism connecting them has no rational homology in dimension 2.) Let JZ = m〈−1〉 ⊕ JZ , where JZ has no elements of square −1. Note that