The authors study some algebraic topological properties (cohomology, Massey products, formal minimal model, rational homotopy theory, fundamental group) of Sasakian manifolds. This study starts from problems presented in the book [Sasakian geometry. Oxford: Oxford University Press (2008; Zbl 1155.53002) (Ch.7)] by C. P. Boyer and K. Galicki. The problems are the following:

(1)

Are there obstructions to the existence of Sasakian structures expressed in terms of Massey products?

(2)

There are obstructions to the existence of Sasakian structures in terms of Massey products, which depend on basic cohomology classes of the related \(K\)-contact structure. Can one obtain a topological characterization of them?

(3)

Do there exist simply connected \(K\)-contact non-Sasakian manifolds (open Problem 7.4.1, [op. cit.])?

(4)

Which finitely presented groups can be realized as fundamental groups of compact Sasakian manifolds?

In the present paper answers to these problems are given.
The main results are the following:
{Theorem 3.2.} For every \(n\geq 3\), there exists a simply connected compact regular Sasakian manifold \(M^{2n+1}\), of dimension \(2n+1\), which is non-formal. More precisely, there is a non-trivial 3-sphere bundle over \((S^2)^{2n-1}\) which is a non-formal simply connected compact regular Sasakian manifold.
{Theorem 4.4.} Let \(M\) be a compact Sasakian manifold. Then, all the higher-order Massey products for \(M\) are zero.
{Theorem 4.9}. Let \(M\) be a simply connected compact symplectic manifold of dimension \(2k\) with an integral symplectic form \(\omega\). Assume that the quadruple Massey product in \(H^{\ast}(M)\) is non-zero. There exists a sphere bundle \(S^{2m-1}\rightarrow E\rightarrow M\), for \(m+1>k\), such that the total space \(E\) is \(K\)-contact, but \(E\) does not admit any Sasakian structure.
{Proposition 5.4}. Let \(\Gamma\) be an irreducible arithmetic lattice in a semisimple real Lie group \(\mathcal{G}\) of rank at least 2 with no co-compact factors and with trivial center. If \(\Gamma\) is Sasakian, then it must be isomorphic to the group \(\pi^{orb}_1(M)\) of some Kähler orbifold. Moreover, \(\Gamma\) cannot be a co-compact arithmetic lattice in \(SO(1,n),n>2\), or \(F_{4(20)}\), or a simple real non-Hermitian Lie group of non-compact type with real rank at least 20.