Constraints for Seiberg-witten Basic Classes of Glued Manifolds

We use rudiments of the Seiberg-Witten gluing theory for trivial circle bundles over a Riemann surface to relate de Seiberg-Witten basic classes of two 4manifolds containing Riemann surfaces of the same genus and self-intersection zero with those of the 4-manifold resulting as a connected sum along the surface. We study examples in which this is enough to describe completely the basic classes. 1. Statement of results Since their introduction nearly a year ago the Seiberg-Witten invariants have proved to be at least as useful as their close relatives the Donaldson invariants. These provide differentiable invariants of a smooth 4-manifold, whose construction is very similar in nature to the Donaldson invariants. Conjecturally, they give the same information about the 4-manifold, but they are much easier to compute in many cases, e.g. algebraic surfaces (see [12]). Problems in 4-dimensional topology are far from solved with these invariants. Nonetheless it is intriguing to compute them for a general 4-manifold. The first step towards it is obviously to relate the invariants of a manifold with those of the manifold which results after some particular surgery on it. Much progress has been made [4] [12]. One natural case to think about is that of connected sum along a codimension 2 submanifold (see Gompf [6]). The typical case would be: Let X̄i be smooth oriented manifolds and let Σ be a Riemann surface of genus g ≥ 1. Suppose we have embeddings Σ →֒ X̄i with image Σi representing a nontorsion element in cohomology whose self-intersection is zero. We form X = X̄1#ΣX̄2 removing tubular neighbourhoods of Σ in both X̄i and gluing the boundaries Y and Y by some diffeomorphism φ. These boundaries are diffeomorphic to Σ × S. The ∗Supported by a grant from Banco de España