Simply-connected irreducible 4-manifolds with no symplectic structures
type: Item
instance of: scholarly article
title: Simply-connected irreducible 4-manifolds with no symplectic structures
zbMATH Open document ID: 0906.57014
author: Zoltán Szabó
published in: Inventiones Mathematicae
publication date: 1999-02-14T00:00:00Z
review text: It is well known that each simply connected smooth 4-manifold \(X\) can be decomposed as \(X=X_1 \# \dots \# X_n\) where all \(X_i\) are irreducible (i.e.: \(X_i=Y_i \# Z_i\) implies either \(Y_i\) or \(Z_i\) to be a homotopy 4-sphere). Moreover, results of \textit{R. E. Gompf} and \textit{T. S. Mrowka} [Ann. Math., II. Ser. 138, No. 1, 61-111 (1993; Zbl 0805.57012)] prove that it is not possible to assume that either \(X_i\) or \(\overline X_i\) have complex structure. On the other hand, the so-called ``Minimal Conjecture'' [\textit{C. H. Taubes}, I. International Press Lectures, UC Irvine (March 1996)] and \textit{D. Kotschick} [Astérisque 241, Exp. No. 812, 195-220 (1997; Zbl 0882.57026)] states that each \(X_i\) may be assumed to be a symplectic 4-manifold with the symplectic and the oppposite orientations allowed. The present paper disproves the above conjecture, by producing -- via logarithmic transformations along embedded tori of self-intersection 0 -- a family of simply connected irreducible 4-manifolds \(X_n\) with odd intersection form so that neither \(X_n\) nor \(\overline X_n\) have symplectic structure. Finally, the construction of counterexamples \(X_n\) is generalized, in order to produce other simply connected irreducible non-symplectic 4-manifolds realizing different homotopy types (including some even intersection forms).
Mathematics Subject Classification ID: 57R57
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R15
zbMATH DE Number: 1186148
zbMATH Keywords: intersection form
zbMATH Keywords: irreducible manifold
zbMATH Keywords: symplectic manifold
zbMATH Keywords: Seiberg-Witten invariant
zbMATH Keywords: smooth 4-manifold
reviewed by: Maria Rita Casali
MaRDI profile type: Publication
full work available at URL: s002220050230
OpenAlex ID: W2006143940
DOI: 10.1007/S002220050230
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description: scientific article; zbMATH DE number 1186148