Osaka Journal of Mathematics

Formality and hard Lefschetz property of aspherical manifolds

Hisashi Kasuya

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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.

Article information

Osaka J. Math., Volume 50, Number 2 (2013), 439-455.

First available in Project Euclid: 21 June 2013

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Zentralblatt MATH identifier

Primary: 20F16: Solvable groups, supersolvable groups [See also 20D10] 55P20: Eilenberg-Mac Lane spaces 55P62: Rational homotopy theory
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 32J27: Compact Kähler manifolds: generalizations, classification


Kasuya, Hisashi. Formality and hard Lefschetz property of aspherical manifolds. Osaka J. Math. 50 (2013), no. 2, 439--455.

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