Extended simplicial rational Nomizu's Theorem

TL;DR

This paper extends a classical theorem to virtually polycyclic groups, providing a new way to compute their cohomology and constructing examples of manifolds with the hard Lefschetz property.

Contribution

It introduces a canonical homomorphism linking finite-dimensional cochain complexes to the de Rham complex of the classifying space, leading to new geometric examples.

Findings

Cohomology isomorphism via the homomorphism

Construction of Sullivan's minimal model for certain groups

New examples of hard Lefschetz symplectic and contact manifolds

Abstract

For a torsion-free virtually polycyclic group $\Gamma$, we give a canonical homomorphism form certain finite-dimensional cochain complex to the $\Q$-polynomial de Rham complex of the simplicial classifying space $B\Gamma$ which induces a cohomology isomorphism. By this result, we obtain the Sullivan's minimal model of certain differential graded algebra defined on $B\Gamma$ and we obtain new examples of hard Lefschetz symplectic manifolds and hard Lefschetz contact manifolds.