Topology of 3-cosymplectic manifolds

TL;DR

This paper investigates the topology of 3-cosymplectic manifolds, revealing Lie algebra actions on their cohomology, establishing topological obstructions, and providing a novel example of such a manifold not decomposable into a product.

Contribution

It introduces new topological bounds on 3-cosymplectic manifolds and presents a nontrivial example that challenges the typical product structure assumption.

Findings

Lie algebra so(4,1) acts on basic cohomology of 3-cosymplectic manifolds

Topological obstructions are expressed via bounds on Betti numbers

Existence of a nontrivial 3-cosymplectic manifold not decomposable into a product

Abstract

We continue the program of Chinea, De Leon and Marrero who studied the topology of cosymplectic manifolds. We study 3-cosymplectic manifolds which are the closest odd-dimensional analogue of hyper-Kaehler structures. We show that there is an action of the Lie algebra so(4,1) on the basic cohomology spaces of a compact 3-cosymplectic manifold with respect to the Reeb foliation. This implies some topological obstructions to the existence of such structures which are expressed by bounds on the Betti numbers. It is known that every 3-cosymplectic manifold is a local Riemannian product of a hyper-Kaehler factor and an abelian three dimensional Lie group. Nevertheless, we present a nontrivial example of compact 3-cosymplectic manifold which is not the global product of a hyper-Kaehler manifold and a flat 3-torus.