Gluing 4-manifolds along \(\Sigma (2,3,11)\)
The authors define a smooth invariant for four manifolds with boundary diffeomorphic to the Seifert fibered homology 3-sphere \(\Sigma(2,3,11)\). Their invariant is based on Seiberg-Witten theory for 4-manifolds with homology 3-sphere boundaries, plus the fact that the three-dimensional Seiberg-Witten equations on \(\Sigma(2,3,11)\) have exactly two non-trivial solutions. They prove the following result, which relates their invariant to the usual Seiberg-Witten invariant: Suppose that a four manifold \(Z\) decomposes as \(Z=X\bigcup_{\Sigma(2,3,11)}Y\), where \(X\) and \(Y\) are four manifolds with boundary \(\pm\Sigma(2,3,11)\). Let \({\mathcal SW}(X)\) and \({\mathcal SW}(Y)\) denote the formal power series formed in the standard way from the invariants defined in this paper, and let \({\mathcal SW}(Z)\) denote the formal power series formed from the Seiberg-Witten invariants of \(Z\). Then \[ {\mathcal SW}(Z)={\mathcal SW}(X)\cdot {\mathcal SW}(Y)\;. \] As an application of this formula the authors prove that there exists a four manifold which contains no embedded Gompf nucleus \(N(2)_{p,q}\).
scientific article; zbMATH DE number 1506031