On some examples in Symplectic Topology

Abstract

Article is devoted to the Examples 2 and 3 of the symplectic solvable Lie groups $R$ with some special cohomological properties, which have been constructed by Benson and Gordon. But they are not succeeded in constructing corresponding compact forms for symplectic structures on these Lie groups. Recently A.Tralle proved that there is no compact form in the Example 3. But his proof is rather complicated and uses some very special topological methods. We propose much more simpler (and purely algebraic) method to prove the main result of the Tralle's paper. Moreover we prove that for Example 2 there is no compact form too. But it appears that some modification of the construction of the Example 2 gives some other example of a solvable Lie group $R^{\prime}$ with the same cohomological properties as $R$, but with a compact form.