Simply-connected irreducible 4-manifolds with no symplectic structures (Q1127750)

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).