Salamon, Dietmar Removable singularities and a vanishing theorem for Seiberg-Witten invariants. (English) Zbl 0865.57019 Turk. J. Math. 20, No. 1, 61-73 (1996). This is an expository paper. The goal is to give a proof of the following vanishing theorem for the Seiberg-Witten invariants of connected sums of smooth 4-manifolds.Theorem 1.1. Suppose that \(X\) is a compact oriented smooth 4-manifold diffeomorphic to a connected sum \(X_1\# X_2\) where \(b^+(X_1)\geq 1\), \(b^+(X_2)\geq 1\), and \(b^+(X)-b_1(X)\) is odd. Then the Seiberg-Witten invariants of \(X\) are all zero.This result is the Seiberg-Witten analogue of Donaldson’s original theorem about the vanishing of the instanton invariants [S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds (1990; Zbl 0820.57002)] for connected sums. An outline of the proof of Theorem 1.1 was given by S. K. Donaldson in [The Seiberg-Witten equations and 4-manifold theory (Preprint, June 1995)]. The key ingredient of the proof is a removable singularity theorem for the Seiberg-Witten equations on flat Euclidean 4-space. A proof of Theorem 1.1 was also indicated by Witten in his lecture on 6 December 1994 at the Isaac Newton Institute in Cambridge. The result was used by D. Kotschick in his proof that (simply connected) symplectic 4-manifolds are irreducible [On irreducible 4-manifolds (Preprint, 1995)]. Cited in 25 Documents MSC: 57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) Keywords:Seiberg-Witten invariants; connected sums; 4-manifolds; removable singularity Citations:Zbl 0820.57002 PDFBibTeX XMLCite \textit{D. Salamon}, Turk. J. Math. 20, No. 1, 61--73 (1996; Zbl 0865.57019)