On formality of Sasakian manifolds

Abstract

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.

Examples of simply-connected K-contact non-Sasakian manifolds of dimension 5

“…There are some well-known topological obstructions to the existence of a Sasakian structure on a compact K-contact manifold (refer to, e.g. [6,2,4,9,1]). …”

Mixed Hodge structures and Sullivan’s minimal models of Sasakian manifolds

“…See [2,14] for the formality of Sasakian manifolds. But we can apply some Hodge theoretical properties of Sasakian manifolds like algebraic varieties.…”

Homotopic Properties of Kähler Orbifolds

“…A simply connected non-formal Sasakian manifold. Examples of simply connected non-formal Sasakian manifolds, of dimension 2n + 1 ≥ 7, are given in [6]. There it is proved that those examples are non-formal because they are not 3-formal, in the sense of Definition 2.2.…”

“…However, opposed to Kähler orbifolds, formality is not an obstruction to the existence of a Sasakian structure even on simply connected manifolds [6]. On the other hand, all quadruple and higher order Massey products are trivial on any Sasakian manifold.…”

“…Clearly, ω is a real closed 2-form on M such that ω 3 > 0, so, ω is a symplectic form on M. Moreover, the form ω is Z 6…”