2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/125352In [HT], two of us constructed a closed oriented 4-dimensional manifold with fundamental group $\Z$ that does not split off $S^1\times S^3$. In this note we show that this 4-manifold, and various others derived from it, do not admit smooth structures. Moreover, we find an infinite family of 4-manifolds with exactly the same properties (and same intersection form on $H_2$). As a corollary, we obtain topologically slice knots that are not smoothly slice in any rational homology ball.We strengthened the statement of Corollary 1.4 and deleted the remark right after Corollary 1.4Geometric Topology57R10 (smoothing)text