Formality and hard Lefschetz property of aspherical manifolds

Abstract

For a Lie group $G = \mathbb{R}^{n}\ltimes_{\phi}\mathbb{R}^{m}$ with the semi-simple action $\phi\colon \mathbb{R}^{n}\to \Aut(\mathbb{R}^{m})$, we show that if $\Gamma$ is a finite extension of a lattice of $G$ then $K(\Gamma, 1)$ is formal. Moreover we show that a compact symplectic aspherical manifold with the fundamental group $\Gamma$ satisfies the hard Lefschetz property. By those results we give many examples of formal solvmanifolds satisfying the hard Lefschetz property but not admitting Kähler structures.