In the present chapter we discuss the analogue of the Lutz—Martinet theorem for simply connected 5—manifolds. Throughout, we assume contact structures ξ to be cooriented, i.e. defined as ξ = ker α by a global 1—form defining the coorientation of ξ. Moreover, if an orientation of the 5—manifold has been chosen, it is understood that the contact structure is positive, that is, α ∧ (dα)2 is a positive volume form. As we saw in Section 2.4, a cooriented contact structure on an oriented manifold M induces an almost contact structure, that is, in the case of 5—manifolds, a reduction of the structure group of the tangent bundle TM from SO(5) to U(2) × 1.

Theorem 8.0.6Every closed, oriented, simply connected 5—manifold admits a contact structure in every homotopy class of almost contact structures.

This theorem was (essentially) proved in [91]. In retrospect I regard my treatment of orientations in that paper as somewhat frivolous; in order to address this issue the present chapter includes a discussion of self-diffeomorphisms of simply connected 5—manifolds.

A word of caution is in order.