Simply connected K-contact and Sasakian manifolds of dimension 7

Abstract

We construct a compact simply connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply connected 7-dimensional Sasakian manifold has vanishing cup product H 2 ×H 2 →H 4 and that it is formal if and only if all its triple Massey products vanish.