arXiv subjects
Publications and source records attributed to G. Placini.
We study fundamental groups of compact Sasaki manifolds and show that compared to K\"ahler groups, they exhibit rather different behaviour. This class of groups is not closed under taking direct products, and there is often an upper bound on the dimension of a Sasaki manifold realising a given group. The richest class of Sasaki groups arises in dimension 5.
In all odd dimensions $\geq 5$ we produce examples of manifolds admitting pairs of Sasaki structures with different basic Hodge numbers. In dimension $5$ we prove more precise results, for example we show that on connected sums of copies of $S^2\times S^3$ the number of Sasaki structures with different basic Hodge numbers within a fixed homotopy class of almost contact structures is unbounded. All the Sasaki structures we consider are negative in the sense that the basic first Chern class is represented by a negative definite form of type $(1,1)$. We also discuss the relation of these results to contact topology.