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Seiberg-Witten invariants on non-symplectic 4-manifolds. (English) Zbl 0882.57013
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 D. Kotschik, J. W. Morgan and 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)].

##### MSC:
 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 57R57 Applications of global analysis to structures on manifolds 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)