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

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 a 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.