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arxiv: 0910.1175 · v4 · pith:V4TXUOB3new · submitted 2009-10-07 · 🧮 math.GT

Formality and hard Lefschetz property of aspherical manifolds

classification 🧮 math.GT
keywords gammahardlefschetzpropertyasphericalformalgroupaction
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For a Lie group $G=\R^{n}\ltimes_{\phi}\R^{m}$ with the semi-simple action $\phi:\R^{n}\to {\rm Aut}(\R^{m})$, 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 K\"ahler structures.

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