  In [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.
