Formality and hard Lefschetz property of aspherical manifolds

2009/10/07 by Kasuya, Hisashi
#20F16 #22E40 #32J27 #55P20 #58A12 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.0910.1175

Abstract

For a Lie group G=\Rⁿ\ltimes_ϕ\R^m with the semi-simple action ϕ:\Rⁿ→ \rm Aut(\R^m), we show that if Γ is a finite extension of a lattice of G then K(Γ, 1) is formal. Moreover we show that a compact symplectic aspherical manifold with the fundamental group Γ 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.

Related

• Geometrical formality of solvmanifolds and solvable Lie type geometries
• Hard Lefschetz Condition on symplectic non-Kähler solvmanifolds
• Symplectic solvmanifolds not satisfying the hard-Lefschetz condition
• Aspherical groups and manifolds with extreme properties
• Remarks on some compact symplectic solvmanifolds