Contact Structures and Periodic Fundamental Groups

This paper is concerned with the existence of contact structures on (connected, closed, orientable) 5-manifolds with certain finite fundamental groups. As such, it constitutes a sequel to [6] (which gave corresponding existence results for highly connected manifolds of arbitrary (odd) dimension and some ad hoc results for finite fundamental groups) and our joint paper [7], where we showed that every 5-manifold M with fundamental group π1(M) = Z2 and universal cover M̃ a spin manifold can be obtained from one of ten ‘model manifolds’ by surgery along a link of 2-spheres and, as an application of this structure theorem, that every manifold of this kind admits a contact structure. In the present paper we combine the ideas of [7] with those of the extensive literature on the existence of positive scalar curvature (psc) metrics – in particular [10, 13, 14, 15] (see also [8] and [16] for more recent surveys on this literature) – to arrive at the following existence result.