Smooth structures on four-manifolds with finite cyclic fundamental groups

For each nonnegative integer m we show that any closed, oriented topological four-manifold with fundamental group Z₄ₘ₊₂ and odd intersection form, with possibly seven exceptions, either admits no smooth structure or admits infinitely many distinct smooth structures up to diffeomorphism. Moreover, we construct infinite families of non-complex irreducible fake projective planes with diverse fundamental groups.