2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62116In this short article we give a criterion whether a given minimal symplectic 4-manifold with $b_{2}^{+}=1$ having a torsion-free canonical class is rational or ruled. As a corollary, we confirm that most of homotopy elliptic surfaces $E(1}_{K}$, K is a fibered knot in $S^3$, constructed by R. Fintushel and R. Stern are minimal symplectic 4-manifolds with $b_{2}^{+}=1$ which do not admit a complex structure.AMS-LaTeX file, 7 pagesGeometric TopologySymplectic Geometry57R17, 57R57text