Seiberg-Witten invariants on non-symplectic 4-manifolds (Q1355883)

The following theorem is proved: Let \(X\) be a nontrivial Seiberg-Witten invariant defined by \(e\in H^2 (X,Z)\) \((b^+_2 (X)>1)\), and let \(N\) be a manifold with negative definite intersection form. If there are even integers \(\lambda_i\), \(i=1,\dots,n\) such that \(4b_1 (N)= 2\lambda_1+ \dots 2 \lambda_n+ \lambda^2_1+ \dots \lambda^2_n\) and the fundamental group of \(N\) has a nontrivial finite quotient, then the connected sum \(X \sharp N\) has a nontrivial Seiberg invariant but does not admit any symplectic structure. This covers a theorem by \textit{D. Kotschik}, \textit{J. W. Morgan} and \textit{C. H. Taubes} in which the hypothesis \(b_1(N) =0\) is used [Math. Res. Lett. 2, No. 2, 119-124 (1995; Zbl 0853.57020)].