Nearly parallel G2-manifolds: formality and associative submanifolds

We construct new examples of non-formal simply connected compact Sasaki-Einstein 7-manifolds. We determine the minimal model of the total space of any fibre bundle over CP2 with fibre S-1 x S-2 or a lens space S-3/Z(p) (p > 0), and we apply this to conclude that the Aloff-Wallach spaces are formal. We also find examples of formal manifolds and non-formal manifolds, which are locally conformal parallel Spin(7)-manifolds. On the other hand, we construct associative minimal submanifolds in the Aloff-Wallach spaces and in any regular Sasaki-Einstein 7-manifold; in particular, in the space Q(1, 1, 1) = SU(2) x SU(2) x SU(2)) / U(1) x U(1)) with the natural S-1-family of nearly parallel G2-structures induced by the SasakiEinstein structure. In each of those cases, we obtain a family of non-trivial associative deformations.