The Seiberg-Witten invariants and symplectic forms

(Note: There are no symplectic forms on X unless b and the first Betti number of X have opposite parity.) In a subsequent article with joint authors, a vanishing theorem will be proved for the Seiberg-Witten invariants of a manifold X, as in the theorem, which can be split by an embedded 3-sphere as X−∪X+ where neither X− nor X+ have negative definite intersection forms. Thus, no such manifold admits a symplectic form. That is,