Some Examples of Aspherical Symplectic Four-manifolds

We give examples of aspherical symplectic 4-manifolds, which are formal in the sense of rational homotopy theory but have degenerate Lefschetz pairings. (preliminary version) Compact Kähler manifolds have a number of topological properties which are not shared by general symplectic manifolds, see [1] for example. In particular, Kähler manifolds have the following cohomological properties: 1. they have even Betti numbers in odd degree, 2. the hard Lefschetz property, and 3. formal cohomology algebras, implying in particular the vanishing of all the Massey products. It is natural to wonder to what extent these properties are dependent on each other when some do hold on a symplectic manifold. There is only one obvious implication, which is that the hard Lefschetz property implies the evenness of b1. Beyond that, little is known, according to the recent preprint [3]. In fact, the literature seems to contain no example of a compact symplectic manifold with odd first Betti number and trivial Massey products, and no (symplectically) aspherical example with even Betti numbers, trivial Massey products, but for which the hard Lefschetz property fails. In this note we give examples for both these phenomena among aspherical symplectic 4-manifolds. Our examples are products of the circle with hyperbolic 3-manifolds which fiber over the circle. This class of manifolds also furnishes other interesting examples beyond the above discussion. Let M be any closed oriented 3-manifold which fibers over S with fiber F and monodromy diffeomorphism φ : F → F . By Moser’s Lemma, we may assume that φ preserves a volume form ǫ on F , so Date: March 15, 2008. 1991 Mathematics Subject Classification. 53C15,57M50. The first author is supported by the Deutsche Forschungsgemeinschaft.