The geometry and topology of 3-Sasakian manifolds.

Abstract

This thesis is concerned with the topology and geometry of Sasakian manifolds. Sasaki structures consist of certain contact forms equipped with special Riemannian metrics. Sasakian manifolds relate to arbitrary contact manifolds as Kählerian or projective complex manifolds relate to arbitrary symplectic manifolds. Therefore, Sasakian manifolds are the odd-dimensional analogs of Kähler manifolds. In the first part of the thesis we discuss some geometric invariants of Sasaki structures. Specifically, the socalled basic Hodge numbers, the type and their relation to the underlying contact and almost contact structures are discussed. We produce many pairs of negative Sasakian structures with distinct basic Hodge numbers on the same differentiable manifold in any odd dimension larger than 3. In the second part of the thesis we discuss topological properties of Sasakian manifolds, focussing particularly on the fundamental groups of compact Sasakian manifolds. In parallel with the theory of Kähler and projective groups, we call these groups Sasaki groups. We prove that any projective group is realizable as the fundamental group of a compact Sasakian manifold in every odd dimension larger than three. Similarly, every finitely presentable group is realizable as the fundamental group of a compact K-contact manifold in every odd dimension larger than three. Nevertheless, Sasaki groups satisfy some very strong constraints, some of which are reminiscent of known constraints on Kähler groups. We show that the class of Sasaki groups is not closed under direct products and that there exist Sasaki groups that cannot be realized in arbitrarily large dimension. We prove that Sasaki groups behave similarly to Kähler groups regarding their relation to 3-manifold groups and to free products.