On the realization of symplectic algebras and rational homotopy types by closed symplectic manifolds

TL;DR

This paper investigates which symplectic algebras can be realized by closed symplectic manifolds in even dimensions, providing answers to longstanding questions and linking algebraic conditions to geometric structures.

Contribution

It resolves open questions about the realizability of symplectic algebras and the implications of rational homotopy models for the existence of symplectic structures.

Findings

Symplectic algebras are realizable in all even dimensions divisible by four.

Algebraic conditions on rational homotopy models can imply the existence of symplectic structures.

Addresses questions of Oprea-Tralle and Lupton-Oprea in dimensions six and higher.

Abstract

We answer a question of Oprea-Tralle on the realizability of symplectic algebras by symplectic manifolds in dimensions divisible by four, along with a question of Lupton-Oprea in all even dimensions. This will also allow us to address, in all even dimensions six and higher, another question of Oprea-Tralle on the possibility of algebraic conditions on the rational homotopy minimal model of a closed smooth manifold implying the existence of a symplectic structure on the manifold.