Formality and the Lefschetz property in symplectic and cosymplectic geometry

TL;DR

This paper reviews topological properties like formality and Lefschetz property in symplectic and cosymplectic manifolds, highlighting differences based on connectivity and Betti number parity.

Contribution

It provides a comparative analysis of topological features in Kähler, symplectic, coKähler, and cosymplectic manifolds, emphasizing the role of fundamental group and Betti numbers.

Findings

Distinct topological properties in simply-connected and $b_1=1$ cases.

Differences in formality and Lefschetz property among manifold types.

Insights into parity of Betti numbers in various geometries.

Abstract

We review topological properties of K\"ahler and symplectic manifolds, and of their odd-dimensional counterparts, coK\"ahler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the K\"ahler/symplectic situation) and the $b_1=1$ case (in the coK\"ahler/cosymplectic situation).