Sasaki structures distinguished by their basic Hodge numbers

TL;DR

This paper constructs examples of manifolds with multiple Sasaki structures distinguished by their basic Hodge numbers, especially in dimension 5, revealing unbounded diversity within certain topological classes.

Contribution

It provides the first examples of manifolds with pairs of Sasaki structures having different basic Hodge numbers in all odd dimensions ≥5, with detailed results in dimension 5.

Findings

Existence of pairs of Sasaki structures with different basic Hodge numbers in all odd dimensions ≥5

Unbounded number of such structures on connected sums of S^2×S^3 in dimension 5

All considered Sasaki structures are negative, with negative definite basic first Chern class

Abstract

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.