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Removable singularities and a vanishing theorem for Seiberg-Witten invariants. (English) Zbl 0865.57019

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)].

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)

Citations:

Zbl 0820.57002
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