No abstract

Kollár has found subtle obstructions to the existence of Sasakian structures on 5-dimensional manifolds. In the present article we develop methods of using these obstructions to distinguish K-contact manifolds from Sasakian ones. In particular, we find the first example of a closed 5-manifold M with H 1 (M, Z) = 0 which is K-contact but which carries no semi-regular Sasakian structures.2010 Mathematics Subject Classification. 53C25, 53D35, 57R17, 14J25.

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.

We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using this, we produce a method of constructing simply connected K‐contact non‐Sasakian manifolds. On the other hand, for every n⩾3, we exhibit the first examples of simply connected compact Sasakian manifolds of dimension 2n+1 that are non‐formal. They are non‐formal because they have a non‐zero triple Massey product. We also prove that arithmetic lattices in some simple Lie groups cannot be the fundamental group of a compact Sasakian manifold.

Topology and its Applications 127 (2003) 33-45. doi:10.1016/S0166-8641(02)00041-XReceived by publisher: 2000-08-30Harvest Date: 2016-01-04 12:21:28DOI: 10.1016/S0166-8641(02)00041-XPage Range: 33-4