Four-manifolds without symplectic structures but with nontrivial Seiberg-Witten invariants
The authors give interesting examples of a closed, oriented, 4-manifold which has nontrivial Seiberg-Witten invariant but does not have a symplectic structure. Earlier in his work [ibid. 1, 809-822 (1994; Zbl 0853.57019), see the review above], the third author proved that every closed symplectic manifold has a nontrivial Seiberg-Witten invariant. Hence these examples of the authors can be regarded as counterexamples to any attempt of establishing the converse of the third author's theorem. On the way of constructing these counterexamples, the authors establish two propositions which are of independent interest. The first is on the decomposition of a symplectic 4-manifold. Let \(X\) be a closed, symplectic 4-manifold which decomposes as a smooth connected sum \(Y\# N\). Then one of the summands \(N\) has a negative definite intersection form and its fundamental group has no nontrivial finite quotients. In the setting of Kähler manifold, this assertion had already been established by the aforementioned paper of the third author. The second is on the nontriviality of Seiberg-Witten invariant under the connected sum operation. Let \(Y\) and \(N\) be a closed oriented 4-manifold. If \(Y\) has a nontrivial Seiberg-Witten invariant and if \(b_1 (N)= b^+_2 (N) =0\), then \(Y\# N\) has a nontrivial Seiberg-Witten invariant. Using these two different properties of the connected sum, the authors obtain their examples by letting \(Y\) be as above and \(N\) be a 4-manifold obtained by doing surgery on the \(S^1\)-factor of \(S^1 \times M\), where \(M\) is a rational homology 3-sphere and its fundamental group has a nontrivial finite quotient.
scientific article; zbMATH DE number 817740