Essential manifolds with extra structures

TL;DR

This paper investigates the existence of essential manifolds with extra structures across various classes, establishing implications between their dimensions and demonstrating the strictness of class inclusions.

Contribution

It proves that the existence of certain symplectically aspherical essential manifolds in higher dimensions implies their existence in lower dimensions and shows that the classes of manifolds with these structures are strictly nested.

Findings

Existence in dimension 2n implies existence in dimension 2n-2 for symplectically aspherical manifolds.

All class inclusions are proper, indicating distinct levels of manifold structures.

Results apply to algebraic, symplectic, hard Lefschetz, and cohomologically symplectic manifolds.

Abstract

We consider classes of algebraic manifolds $\mathcal{A}$, of symplectic manifolds $\mathcal{S}$, of symplectic manifolds with the hard Lefschetz property $\mathcal{HS}$ and the class of cohomologically symplectic manifolds $\mathcal{CS}$. For every class of manifolds $\mathcal{C}$ we denote by $\mathcal{EC}(\pi,n)$ a subclass of $n$-dimensional essential manifolds with fundamental group $\pi$. In this paper we prove that for all the above classes with symplectically aspherical form the condition $\mathcal{EC}(\pi,2n)\ne \emptyset$ implies that $\mathcal{EC}(\pi,2n-2)\ne\emptyset $ for every $n>2$. Also we prove that all the inclusions $\mathcal{EA}\subset\mathcal{EHS}\subset\mathcal{ES}\subset\mathcal{ECS}$ are proper.