On certain geometric and homotopy properties of closed symplectic manifolds

TL;DR

This paper explores the relationships between the Hard Lefschetz property, Massey products, and Betti numbers in closed symplectic manifolds, aiming to deepen understanding of their topological and homotopy characteristics.

Contribution

It provides a comprehensive analysis of how these properties interrelate in symplectic manifolds, including summarized tables and discussion on harmonic Betti number variations.

Findings

Relations between homotopy properties are summarized in tables.

Symplectic manifolds can violate classical properties known for Kähler manifolds.

Variation of harmonic Betti numbers in 6-dimensional cases is discussed.

Abstract

The paper deals with relations between the Hard Lefschetz property, (non)vanishing of Massey products and the evenness of odd-degree Betti numbers of closed symplectic manifolds. It is known that closed symplectic manifolds can violate all these properties (in contrast with the case of Kaehler manifolds). However, the relations between such homotopy properties seem to be not analyzed. This analysis may shed a new light on topology of symplectic manifolds. In the paper, we summarize our knowledge in tables (different in the simply-connected and in symplectically aspherical cases). Also, we discuss the variation of symplectically harmonic Betti numbers on some 6-dimensional manifolds.