Minimal models, formality and hard Lefschetz properties of solvmanifolds with local systems

TL;DR

This paper extends the understanding of differential forms on solvmanifolds with local systems by constructing minimal models, refining formality and Lefschetz properties beyond nilmanifolds.

Contribution

It constructs Sullivan's minimal models for differential forms with local systems on solvmanifolds, generalizing Nomizu's theorem and refining existing results on formality and Lefschetz properties.

Findings

Extended Nomizu's theorem to solvmanifolds with local systems

Refined conditions for formality of solvmanifolds

Established hard Lefschetz properties for a broader class of manifolds

Abstract

For a simply connected solvable Lie group G with a cocompact discrete subgroup {\Gamma}, we consider the space of differential forms on the solvmanifold G/{\Gamma} with values in certain flat bundle so that this space has a structure of a differential graded algebra(DGA). We construct Sullivan's minimal model of this DGA. This result is an extension of Nomizu's theorem for ordinary coefficients in the nilpotent case. By using this result, we refine Hasegawa's result of formality of nilmanifolds and Benson-Gordon's result of hard Lefschetz properties of nilmanifolds.